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1 31 st July, 014 Ms. Norah Barger and Mr. Karl Cordewener, Co-Chairs, Trading Book Group Basel Committee on Banking Supervision Bank for International Settlements Centralahnplatz, CH-400 Basel, SWITZERLAND Sent y to: norah.arger@fr.gov; karl.ordewener@is.org; Juquan.Tan@is.org; aselommittee@is.org Re: Seond Consultative Doument Fundamental Review of the Trading Book 1 (CP) BCBS 65 Quantitative Impat Study instrutions Supplementary note on SBA Dear Ms. Barger and Mr. Cordewener, This letter ontains additional omments from the Assoiations [1] on the draft instrutions reeived on 3 rd June 014 in relation to the quantitative impat study ( QIS ), launhed y the Basel Committee on Banking Supervision ( BCBS ) on the Fundamental Review of the Trading Book ( FRTB ). As previously ommuniated in the relevant response (dated 17 th July 014) the industry has identified a flaw in the presried delta risk aggregation formula. Having performed additional analysis, partiipants were ale to identify a numer of potential solutions that an mitigate the ourrene of negative variane in the aggregation formula of delta risk apital harges aross ukets. Herein, our memers outline these preliminary solutions and their pros and ons. Considering the importane of the SBA alulation in the market risk apital framework and the daily alulation of the SBA apital harge over onstantly hanging portfolios one the FRTB framework is implemented, the Assoiations strongly advoate that this issue needs to e addressed and we would e happy to failitate a disussion with the industry experts group in short notie. We stress again our ommitment to partiipate onstrutively in the onsultative proess and we do sinerely hope you find our input helpful. Yours faithfully, 1 Basel Committee on Banking Supervision, Otoer 013 [1] The International Swaps and Derivatives Assoiation, In. ( ISDA ), the Gloal Finanial Markets Assoiation ( GFMA ) and the Institute of International Finane ( IIF )

2 Mark Gheerrant Head of Risk and Capital ISDA David Strongin Exeutive Diretor GFMA Andres Portilla Diretor, Regulatory Affairs IIF.. Ju Quan Tan, Memer of Seretariat, Basel Committee

3 I. INTRODUCTION The FRTB s standardised SBA presries (in paragraph 8) the following method for alulating the Delta Risk apital requirement from weighted sensitivities. First the weighted sensitivities within a given uket are omined to give the apital harge for that uket: Then the apital harges for the various ukets are aggregated as follows: The seond formula has the prolem that the term inside the square root an go negative. We provide an example in appendix 1 that illustrates the ourrene of negative variane in the SBA risk aggregation formula. II. PROBLEM DEFINITION The industry fully appreiates the TBG s effort to make the SBA more risk sensitive, in partiular y apturing hedging and diversifiation enefits through the introdution of a variane-ovariane type of an approah for the aggregation of the apital harges for the various ukets. At the same time we omprehend the diffiulty whih the regulators are faing in i) speifying a large sale orrelation matrix aross all risk fators and ii) estalishing a onvenient and transparent asading approah to the aggregation of apital aross ukets. The proposed methodology adopts a asading approah with the aggregation of apital harges at the risk fator level, then at roader uket level and finally at the asset lass level. This approah has the enefit of requiring only the speifiation of a manageale numer of orrelations in eah step. Moreover, the proposed regulatory formulas define two uket level metris (K and S) that have the following finanial interpretation: K measures uket level risk while apturing orrelation and diversifiation aross risk fators within the same uket; S measures uket level risk assuming no diversifiation aross risk fators of the same uket. A valid K requires that the two orrelation matries (for same signs and different signs) meet the onditions for positive semi-definiteness as illustrated in Appendix 5. The issue with the seond formula is more omplex and is the sujet of the rest of this doument. 3

4 When testing the presried formula, we identified inonsistenies in the outputs regarding apital numers. We identified that this was aused y the use of K and S in the same variane-ovariane-like aggregation formula for asset lass level delta-risk harge. The soure of the issue is that the risk orrelations aross ukets depend on the omposition and partiularly the diretionality of the risk fator exposures within eah uket. Considering a simple example of only two ukets and only one risk fator within eah uket, the sign of the orrelation etween the risky P&L orresponding to the two ukets would depend on whether we are long oth ukets or long one and short the other. This diretional information is lost in the alulation of the metri K whih is always positive if the orrelations meet the onditions illustrated in appendix 5. As suh, speifying the same orrelations for the K s without further information aout the omposition/diretionality of exposures within eah uket an lead to signifiant misrepresentation of asset lass level risk and therefore to the apital alloation. FRTB CP attempts to solve this issue y using K for the uket level variane term while using S for the ross terms. The S eing the sum of risk weighted exposures, whih ontains some information aout diretionality. This however reates another prolem in the formula as the S terms represent uket level risk assuming no risk fator level diversifiation, while the K s represent uket level risk with risk fator level diversifiation. Consequently, there are situations wherey: The ross terms will dominate the diret variane terms and Therey lead to a negative asset lass level variane, Hene giving rise to a square-root of a negative variane during the alulation of delta risk harge at the asset lass level. The industry provided an example of this in our reent ommuniation with the TBG (see Appendix 1). Essentially, the aim of the example was to demonstrate that while it may seem easier and more transparent to use a asading approah for speifying orrelations, one annot freely selet orrelations and use any variane-ovariane-like aggregation. The implied orrelations at the lowest risk fator level an e inonsistent and thus ould lead to eonomially non-meaningful results, suh as negative variane at the asset lass level. In order to avoid delays in the QIS, the industry has disussed a numer of possile solutions to address this prolem in the aggregation of apital harges for SBA, whih we hope that the TBG will onsider in its delierations. In this ontext, we would like to note that given the very short time over whih our memers have put the proposals forward, we have not had the time to onlude on a preferred industry solution at this stage. We would like to disuss the options with the TBG in order to identify potential onsiderations and ome to a onlusion regarding the most suitale option, given the ojetives of the FRTB. 4

5 III. POSSIBLE SOLUTIONS The possile solutions to address the ourrene of negative variane in the SBA apital alulation fall under four different approahes: (1) Diretly deal with the negative variane symptom y apping/flooring the apital harge alulation. Two solutions fall under this line of thought: a. Floor the term inside the square-root of the delta harge equation to 0. This is a rute fore approah; it is more onsistent with the urrent FRTB speifiation and always solves the negative variane prolem. However, it does not deal with the root ause of the inonsisteny issue and we still have two separate uket level metris (K and S) in the same equation.. Cap the asolute value of S at K. This method solves the prolem as long as the underlying orrelation matrix (the rhos) is positive semi-definite. Similar to (a), it does not attempt to deal with the root inonsisteny prolem ut it essentially imposes the onstraint that uket level risk with no diversifiation is smaller than uket level risk with diversifiation. () Eliminate the two uket level metris a. Use signed K instead of S in the delta risk harge equation. The methodology then redues to a standard variane-ovariane form and as long as the orrelation matrix is positive semidefinite, there will not e a negative variane prolem. Please find further analysis on options (1a), (1) and () in Appendies and 3. (3) Redue the magnitude of the ross terms a. The simplest way is to ignore the orrelation aross risk ukets i.e. assume zero orrelation. While this an ertainly eliminate the negative variane prolem, it also disallows any hedging enefits aross ukets, ontraditing the intention for a more risk sensitive standardized approah.. Make the ross ukets orrelations (gamma_ s) smaller ut not zero or set them in a way (e.g. outer orrelation equal to the produt of orresponding inner orrelations) to mitigate the prolem. (4) Instead of using ross terms suh as S*S, use ross terms that are onsistent with orrelation assumptions in the alulation of K a. Together with a saling fator etween 0 and 1 of ross uket orrelation, this an mitigate the negative variane prolem. The downside of this approah is that the ross terms should e alulated over all pairs of ukets and it s not modular as the other options. See Appendix 4 for further desription of this approah. Moreover the fat that FRTB CP speifies different orrelations when exposures have the same sign versus exposures with different signs results to magnifiation of the prolem of ensuring a positive definite orrelation matrix and hinders the effiay of the solutions presented aove. See Appendix 5 for a ondition the same and different sign orrelations have to satisfy in order to ensure the positive definiteness 5

6 IV. CONCLUSION Considering the aove and that the QIS is to start imminently, we onlude that there is not adequate time to work on re-speifiation/realiration of the orrelations aross the different risk fators. Therefore, the only options we present that an e inorporated to immediate QISsthe urrent QIS (ut an e amended in the susequent QISs), would e the simplest ones suh as (1) and () and (3a) whih do not attempt to solve the root ause of the prolem ut do avoid the negative variane issue. Elaorating further on these potential solutions (1a, 1, a and 3a), option 1a is the most onsistent with the FRTB CP/SBA equation. Option 1a simply floors the term within the square-root while options 1 and a effetively alter the SBA equation. Option 1 uses a formula that produes a smoother and ontinuous apital estimate, while option a avoids having two uket level metris and therefore is losest to a standard variane-ovariane formula. Option 3a disallows ross-uket orrelation so is less risk sensitive, ut represents a very simple way to omine ukets. Considering the low level of gammas (ross ukets orrelations) speified in FRTB/ CP, the effet of ignoring ross ukets hedging ould e relatively immaterial although more analysis is needed to validate this point. Considering the limited time that was availale for the analysis of this prolem and the investigation of potential solutions, industry partiipants are expressing the need for more time in exploring these options and understanding the full impliations of applying these methodologies for apital alloation and planning purposes. Therefore we strongly reommend that the TBG analyses the prolem in more detail onsidering the signifiane of the alulation in the overall design of the SBA. The industry is happy to engage into onstrutive dialogue with the TBG on this prolem and disuss the potential solutions desried in this initial paper. 6

7 APPENDIX 1: Example of Negative Variane in the Risk Aggregation Formula Suppose we are alulating the Delta Risk Charge for GIRR (General Interest Rate Risk), and that there are just two urrenies, with 10 tenor points in eah. Suppose we have = 100 and = -100 for all tenors, suppose for all tenors and, and suppose. Then we get,,, and the term inside the square root is -80,000. Digging deeper into this example, we an show that y expanding the terms into the lowest risk fator level, the implied orrelation matrix at the risk fator level is not positive semi-definite. In other words, one is not entirely free in speifying orrelation parameters in a asading approah. To see this, note that the formula for the delta risk harge is: [Delta Risk Charge] where K K WSk S WS k k k k lk kl k S WS WS l S Expanding the harge formula, we have [Delta Risk Charge] WS klws, kws, l WS, kws, k l, k k k lk k i.e., the total variane aross all delta's ased on the following orrelations: : orrelation etween two points within the same uket kl : orrelation etween two points from different ukets Taking the example aove, we have the following 0x0 'orrelation' matrix C: 7

8 WS 1,1 WS,1 50% 40% 40% WS 1,10 WS,10 This matrix has a negative eigenvalue (λ=-0.4) where the WS's (i.e. v = [+100,...,+100,-100,..., -100]) in the example is an eigenvetor. The variane orresponding to this eigenvetor is T T v Cv v v 80,000, reovering the result in the example. 8

9 APPENDIX : Casading in SBA - Three Possile Solutions The SBA methodology has taken a asading approah to variane alulation. The main advantages of this approah are: Avoids the need to speify a large ovariane matrix, with unertain minor ross-terms Easier to speify smaller ovariane matries that are positive semi-definite Relatively easy to give explanations of variane ak to individual risk auses helps risks managers and trading staff to understand the drivers of apital. For a asading approah to apture hedging enefits aross ukets, using K whih is like a uket level standard deviation alone is not suffiient. FRTB therefore employed a signed version of uket risk (S) to apture the hedging effet. Below, we show that eside the partiular hoie in FRTB, there are alternative definitions of S that are not exposed to the same negative variane prolem. Three possiilities for the definition of S Suppose we have three tiers of risk fators (i, j, k), with individual risk weights yield urve risk tiers might e (urreny, tenor, su urve), eg (USD, 5y, OIS). We have three alternative definitions for the signed risk, given the risk apital harge, 1(a), Current (a), Signed-K, ( ) 1(), Cap+Floor, ( ( ) ). For example the Casading equations in full At the first tier we define the risk apital and its signed version as ( ) ( ) { ( ( ) ) Here we have shown three possile alternative definitions for S. Then at the seond tier, we define the risk apital as ( ) 9

10 Under option 1(a), the numer under the square root an eome negative. In this ase defined as zero. The signed version of the risk apital is defined as would e ( ) { ( ( ) ) Finally, at the last tier, the risk apital is ( ) and again floor the variane at zero (only if we are using option 1(a)). There is no need to define the signed version now, sine this is the final result. This tiered approah an e easily extended to any level of tiering as required. Three S definitions ompared Option 1(a), Current, with variane floored at zero This feels like ignoring the prolem rather than solving it. Enountering the square root of a negative numer in the apital alulation is surely indiative of a fairly serious prolem. It is less than redile to have an approah that ould output a apital of zero. Furthermore, the Delta Risk Charge formula aove an give rise to an unreasonaly large apital requirement in the ase where the risks have the same sign so. In this ase the formula does not allow a fair diversifiation enefit. Option (a), Signed-K This is a more sophistiated approah whih is similar to one used in an earlier version of the SBA. It represents an improvement on Option 1(a), ut introdues a new prolem, namely that the apital is a disontinuous funtion of the underlying risks. Sine in general, when goes through zero, then will jump etween and. Thus a tiny hange in the risks of the portfolio ould ause a large hange in the apital, whih is learly undesirale. Option 1(), Cap+Floor This is similar in spirit to Option (a), ut avoids the disontinuity prolem, sine this new definition of is a ontinuous funtion of the weighted sensitivities. 10

11 We an see the intuitive ehaviour of three alternatives as simple graphs. Here the graphs show S against in the ase where K =. Current Signed Cap + Floor As an e seen, all of these three methods will work with tiering. Their features an e summarised in this tale: Definition of S Advantages Disadvantages 1(a) Current Simple, ontinuous Unounded, redues diversifiation enefit. Possile negative variane, whih is ured y flooring the variane at zero. (a) Signed-K 1() Cap+Floor Bounded, always non-negative variane. Continuous, ounded, always non-negative variane. Disontinuous apital ould move signifiantly due to tiny hange in risk. More omplex formula. 11

12 (Delta Charge) : with fixes (Delta Charge) : with fixes APPENDIX 3: Further Analysis on Possile Solutions (1a) and 1() The purpose of this analysis is an attempt to understand the possile solutions (1.a) and (1.) in the main text of this doument. We keep the same setting as in the example in the appendix, namely, two ukets ten points per uket ij : 40% However, we onsider two uket-wise orrelations = 30%: it an e shown that this leads to a legitimate orrelation struture. = 50%: as shown in our previous note, this leads to an illegitimate orrelation struture. We randomly generates the following three different types of the WS k 's, i.e. vetors of 0 elements 1. random signs: [x 1,..., x 10, x 11,..., x 0 ] where eah x j is sampled using the standard normal.. all positive signs: [ x 1,..., x 10, x 11,..., x 0 ] 3. opposite signed etween ukets: [+ x 1,..., + x 10, - x 11,..., - x 0 ] The third type is motivated y the fat that the eigenvetor assoiated with negative eigenvalue with = 50% is [+1,..., +1,-1,... -1]. For eah omination, we alulate the square of the delta harges as follows: speified in FRTB in the ways speified in the potential solution (1.a): simple flooring. (1.): aps/floors 1000 simulations are run. 150 = 30% (1.): random signs (1.): same signs (1.): opposite signs (1.a) 150 = 50% (1.): random signs (1.): same signs (1.): opposite signs (1.a) (Delta Charge) : as speified (Delta Charge) : as speified 1

13 Oservations: Solution (1.a): With = 50%, this approah an generate negative values and should e floored. Otherwise, the results are idential to the standard varianes. Solution (1.): Even with = 30%, where the orrelation is legitimate, the approah gives different values from the standard variane. It an yield values systemially higher (opposite sign ase) or lower (same sign ase) than the standard varianes. Fousing on the ase of = 50% and opposite signs, there seems to e no lear relationship with the standard varianes. Therefore, it seems to require further analysis to understand the approah. 13

14 APPENDIX 4: Possile Solution (4) Single Correlations Saled y γ As disussed in Appendix 1, the root ause of the prolem of having a negative variane is a misspeifiation of the orrelation struture. As having negative eigenvalues an potentially lead to lower apital harges, it is desirale to speify legitimate orrelation strutures. Nonetheless, it may not e an easy task to prove that the orretions are orretly speified, i.e. positive semi-definite. Having a single orrelation etween two different ukets would not e ideal, e.g. USD 1Y and EUR 1Y would e more orrelated than USD 1Y and EUR 30Y. This solution orrets the fundamental prolem that inonsistent orrelations an result in negative variane. It is a simple way of speifying the orrelations etween risks in different urrenies/ukets in suh a way that gloal orrelation matrix is self-onsistent (ie. positive semi-definite). No additional inputs to those that are urrently used. Correlations are derived using the existing single urreny orrelation matrix and the γ values speified in the rule. Compatile with solution (1.a) as a failsafe in ase modifiations to the alulation introdue the possiility of negative variane. For revity, we refer to risk for a urreny. This an e generalized to urreny, urve, or uket as appropriate. Similarly, exposures y maturity an e generalized to exposures y produt, redit lass, et. Re-ap of Current Method: The urrent method uses the following formulas: (1) S i WS, i () [Delta Risk Charge] K SS For larity, we define a value X whih is the ross variane etween urrenies. Using X in the urrent method, we get (3) X, SS Note that this is equivalent to X, WS, iws, j (4) [Delta Risk Charge] K X i j Solution 4a: We propose hanging the formula for X to use the single urreny orrelations in the alulation. 14

15 (3a) X, ijws, iws, i j Then use (4) to alulate the delta risk harge. j Eonomially, this has a simple interpretation. The orrelations etween urrenies are simply a saled down version of the orrelations within a urreny. γ is the sale fator and an range from 0 to 1. For example, suppose γ = 0.5 and we have exposures in the 10y uket. If they are the same urreny, the orrelation would e 1.0. If they are in different urrenies, the orrelation is 0.5 * 1.0 = 0.5. If we had same sign exposures in the 5y and 10y uket, the single urreny orrelation is 0.9. For exposures in different urrenies the orrelation would e On Validity of Proposal: If the single urreny orrelation matrix is valid, the proposed method always results in a non-negative variane: Let C e a valid orrelation matrix for a single urreny. x and y will e vetors for risk weighted exposures. (a) C is positive semi-definite, so for all vetors of real numers x: x T Cx 0 Let x and y represent portfolios in different urrenies. The variane of x+y will e T () ( x y) C( x y) 0 The variane an e rewritten as follows T ( x y) C( x y) T T T T x Cx x Cy y Cx y Cy T T T x Cx y Cy * y Cx Putting the new form into () we get T T T () x Cx y Cy * y Cx 0 The 3 terms of the variane of x+y are the variane of x, the variane of y, and the ontriution of the ross orrelation etween x and y. The first terms are the K terms in (4). We know that this quantity is non-negative. T T x Cx y Cy K The third term is the ross urreny exposure and orresponds to the seond part in (4), ut without the sale fator. T * y Cx X Sine the first terms are non-negative and the total is non-negative, multiplying the third term y a onstant, γ, etween 0 and 1 must result in a non-negative sum. T x Cx y T Cy * y T Cx K X 0 15

16 If γ is the same for all urreny pairs, whih is the urrent rule, the result applies to any numer of urrenies. We reommend that the ommittee keep this speifiation. Notes: If the ommittee hooses to speify different γ for different urreny pairs we annot rule out the possiility of negative variane. In partiular, it is possile to speify inonsistent values for triplets of urrenies. The only solution would e a validation of the full orrelation matrix. Taking into aount the same- and different-sign orrelations, the orrelation strutures C + = + ij and C - = - ij over all maturities aross the ukets would look like the followings: ρ + ij γ ρ + ij γ ρ + ij ρ - ij γ ρ - ij γ ρ - ij γ ρ + ij ρ + ij v v γ ρ + ij ρ - ij v γ ρ + ij γ ρ - ij γ ρ + ij γ ρ + ij ρ + ij γ ρ - ij γ ρ - ij ρ - ij o o Then, ij ij 0 ij and ij satisfy the onditions in Appendix 5 if ij One of and ij is positive definite. On Casading: o In this approah, the strit uket-y-uket 'asading' is lost: (3a) should e alulated aross all pairs. o On the other hand, the delta harge is an aggregated sum of the following extended ross-asading struture K 's and X 's: K Value X GBP EUR USD JPY GBP 34 GBP 4 3 EUR 13 EUR -3 - USD 43 USD 1 JPY JPY 16

17 o In ase of three or more asading, the approah an e rather omplex than other asading approahes in Appendix 4. 17

18 APPENDIX 5: Same and Different Sign Correlations The purpose of this setion is to present a ondition where the same- and different sign orrelations to satisfy in order to ensure the positive definiteness. Let ij and Assume that ij e the orrelations for the pairs of the same and different signs, respetively. 0 ij One of Then, we have t v Cv if ij ij i v v 0 i i ij and v v j ij is positive definite. ij i v v 0 i i v v j v v ij i j 0 ij is positive definite. A similar proof an e made if ij is positive definite. 18