SBA Risk Measures

SBA Risk Measures

Relevant provisions: paragraphs 47 (c), 67 and 71 of the January 2016 market risk framework.

BCBS response: “Yes, as per paragraph 47 (c), a bank may make use of alternative formulations of sensitivities based on pricing models that the bank’s independent risk control unit uses to report market risks or actual profits and losses to senior management. In doing so, the bank is to demonstrate to its supervisor that the alternative formulations of sensitivities yield results very close to the prescribed formulations.”

Referenced FRTB text:

Questions related to SBA, delta sensitivity (1.1) reference paragraphs 47 (c), 67 and 71 of the January 2016 market risk framework.

47 (c) The bank must determine each delta and vega sensitivity and curvature scenario based on instrument prices or pricing models that
an independent risk control unit within a bank uses to report market risks or actual profits and losses to senior management.

67. Sensitivities for each risk class are expressed in the reporting currency of the bank.

67(a) Delta GIRR: Sensitivity is defined as the PV01 (sensitivity) of an instrument i with respect to vertex t of the risk-
free yield curve (or curves, as appropriate) used to price the instrument i for the currency in which i is denominated. PV01 is determined by calculating the change in the market value of the instrument (Vi (.)) as a result of a 1 basis point shift in the interest rate r at vertex t (rt) of the risk-free yield curve in a given currency, divided by 0.0001 (ie, 0.01%). In notation form:

where:
rt is the risk-free yield curve at vertex t;
cst is the credit spread curve at vertex t;
Vi (.) is the market value of the instrument i as a function of the risk-free interest rate curve and credit spread curve.

67(b) Delta CSR non-securitisation:
Sensitivity is defined as CS01.The CS01 (sensitivity) of an instrument i is determined by calculating the change in the market value of the instrument (Vi (.)) as a result of a 1 basis point change to credit spread cs at vertex t (cst), divided by 0.0001 (ie, 0.01%). In notation form:

67(c) Delta CSR securitisation and nth-to-default:
Sensitivity is defined as the CS01, with no change to the sensitivity specification in the previous paragraph.

67(d) Delta equity spot: The sensitivity is calculated by taking the value of a one percentage point change in equity spot price, divided
by 0.01 (ie, 1%). In notation form:

where:
k is a given equity;
EQk is the market value of equity k; and
■ Vi (.) is the market value of instrument i as a function of the price of equity k.

67(e) Delta equity repos: The sensitivity is calculated by taking the value of a one basis point absolute translation of the equity repo rate term structure, divided by 0.0001 (ie, 0.01%). In notation form:

where:
k is a given equity;
RTSk is the repo term structure of equity k; and
Vi (.) is the market value of instrument i as a function of the repo term
structure of equity k.

67(f) Delta commodity: The sensitivity is calculated by taking the value of a one percentage point change in commodity spot price, divided by 0.01 (ie, 1%):

where:
k is a given commodity;
CTYk is the market value of commodity k; and
Vi (.) is the market value of instrument i as a function of the spot price
of commodity k.

67(g) Delta FX: The sensitivity is calculated by taking the value of a one percentage point change in exchange rate, divided by 0.01 (ie,1%):

where:
k is a given currency;
FXk is the exchange rate between currency k and the reporting
currency; and
Vi (.) is the market value of instrument i as a function of the exchange rate k.

Authors’ comment: In its response, BCBS clarifies that it allows banks to utilise their existing pricing models to formulate delta sensitivities if they are approved by the independent risk control unit and the supervisor.

Relevant provisions: Paragraph 49 (a) of the January 2016 market risk framework.

BCBS response: Yes, only instruments with non-zero vega sensitivities are subject to vega and curvature risk charges. In the example cited, CMS are subject to vega and curvature risk charges.

Referenced FRTB text:
49 (a) Each instrument with optionality is subject to vega risk and curvature risk. Instruments without optionality are not subject to
vega risk and curvature risk.

Authors’ comment: Trivially, vega and curvature risk charges need only be calculated on instruments that have optionality – ie, instruments that generate vega and curvature. Although the example cited clearly generates both vega and curvature, vega and curvature calculations would not be required on plain vanilla instruments such as cash bonds or equity.

Relevant provisions: paragraph 52 of the January 2016 market risk framework

BCBS response: The delta used for the calculation of the curvature risk charge should be the same as that used for calculating the delta risk charge. The assumptions that are used for the calculation of the delta (ie, sticky delta for normal or log-normal volatilities) should also be used for the calculation of the shifted or shocked price of the instrument.

Referenced FRTB text:
(v) Sensitivities-based method: curvature

52. The curvature risk charge consists of a set of stress scenario on given risk factors, which are defined in Section 3. Two stress scenarios
are to be computed per risk factor (an upward shock and a downward shock) with the delta effect, already captured by the delta risk
charge, being removed. The two scenarios are shocked by risk weights and the worst loss is aggregated by correlations provided in
Section 6. The purpose of this subsection is to provide the aggregation formulas within buckets, and across buckets within a risk class.

Authors’ comment: This question replaces the 2017 Q3 in section 1.1. In order to calculate shock curvature shock values (up and down) under SBA, the delta risk charge must first be extracted to avoid double counting. There had been some question about whether this extraction process could involve different assumptions for sticky delta. The BCBS response clarifies that all assumptions must be aligned in order to preserve a matching between delta risk charge captured and delta risk charge extracted from curvature.

Relevant provisions: Paragraph 53 of the January 2016 market risk framework.

BCBS response: No such floor is permitted in the market risk standard.

Referenced FRTB text:

53. The following step-by-step approach to capture curvature risk must be separately applied to each risk class (apart from default risk):

(a) Find a net curvature risk charge CVRk across instruments to each curvature risk factor k. For instance, all vertices of all the curves
within a given currency (eg, Euribor three months, Euribor six months, Euribor one year, etc for euro) must be shifted upward. The potential loss, after deduction of the delta risk positions, is the outcome of the first scenario. The same approach must be followed on a downward scenario. The worst loss (expressed as a positive quantity), after deduction of the delta risk position, is the curvature risk position for the considered risk factor. If the price of an option depends on several risk factors, the curvature risk is determined separately for each risk factor.

(b) The curvature risk charge for curvature risk factor k can be formally written as follows:

where:
i is an instrument subject to curvature risks associated with risk factor k;
■ xk is the current level of risk factor k;
■ Vi(xk) is the price of instrument i depending on the current level of risk factor k;
■ – Vi(xk(RW(curvature)+)) and Vi(xk (RW(curvature)–)) both denote the price of instrument i after xk is shifted (ie, “shocked”) upward and downward;
■ under the FX and equity risk classes:
● RWk(curvature) is the risk weight for curvature risk factor k for instrument i determined in accordance with paragraph 131; and
● sik is the delta sensitivity of instrument i with respect to the delta risk factor that corresponds to curvature risk factor k.
■ under the GIRR, CSR and commodity risk classes:
● RWk(curvature)  is the risk weight for curvature risk factor for instrument i determined in accordance with paragraph 132; and
● sik is the sum of delta sensitivities to all tenors of the relevant curve of instrument i with respect to curvature risk factor k.

(c) The aggregation formula for curvature risk distinguishes between positive curvature and negative curvature risk exposures.The negative curvature risk exposures are ignored, unless they hedge a positive curvature risk exposure. If there is a negative net curvature risk exposure from an option exposure, the curvature risk charge is zero.

(d) The curvature risk exposure must be aggregated within each bucket using the corresponding prescribed correlation ρkl as set out in the following formula:

where ψ(CVRk, CVRl) is a function that takes the value 0 if CVRk and CVRl both have negative signs. In all other cases, ψ(CVRk,CVRl) takes the value of 1.

(e) Curvature risk positions must then be aggregated across buckets within each risk class, using the corresponding prescribed correlations
γbc

where:
■ Sb= ΣkCVRk  for all risk factors in bucket b, and Sc = ΣkCVRk in bucket c; and
■ ψ(Sb, Sc) is a function that takes the value 0 if Sb and Sc both have negative signs. In all other cases, ψ(Sb, Sc) takes the value of 1.
If these values for Sb and Sc produce a negative number for the overall sum of ΣbKb 2 + ΣbΣc≠bybcSbScψ (Sb ,Sc):
■ the bank is to calculate the curvature risk charge using an alternative specification whereby Sb = max[min(ΣkCVRk,Kb),–Kb] for all risk factors in bucket b and Sc = max[mm(Σ,k-CVRk,Kc),–Kc]  for all risk factors in bucket c.

Authors’ comment: The concern expressed in this question relates to odd results stemming from negative interest rate environments.BCBS’s response is direct and self-explanatory.

Relevant provisions: Paragraphs 59, 60, 62, 63, 64, 65 and 66 of the January 2016 market risk framework.

BCBS response: Banks are not permitted to perform capital computations based on internally used vertices. Risk factors and sensitivities must be assigned to the prescribed vertices. As stated in footnotes 15, 21 and 22, assignment of risk factors and sensitivities to the specified vertices should be performed by linear interpolation or a method that is most consistent with the pricing functions used by the independent risk control function of the bank to report market risks or profits and losses to senior management.

Referenced FRTB text:

59. General Interest Rate Risk (GIRR) risk factors

(a) Delta GIRR: The GIRR delta risk factors are defined along two dimensions: a risk-free yield curve for each currency in which interest rate-sensitive instruments are denominated and the following vertices: 0.25 years, 0.5 years, one year, two years, three years, five years, 10 years, 15 years, 20 years, 30 years, to which delta risk factors are assigned.3

i. The risk-free yield curve per currency should be constructed using money market instruments held in the trading book which have the lowest credit risk, such as overnight index swaps (OIS). Alternatively, the risk-free yield curve should be based on one or more market-implied swap curves used by the bank to mark positions to market. For example, interbank offered rate (BOR) swap curves.

ii. When data on market-implied swap curves described in (a) (i) is insufficient, the risk-free yield curve may be derived from the most appropriate sovereign bond curve for a given currency. In such cases the sensitivities related to sovereign bonds is not exempt from the credit spread risk charge: when a bank cannot perform the decomposition y = r + cs,any sensitivity of cs to y is allocated to the GIRR and to CSR risk classes as appropriate with the risk factor and sensitivity definitions in the standardised approach. Applying swap curves to bond-derived sensitivities for GIRR will not change the requirement for basis risk to be captured between bond and CDS curves in the CSR risk class.

iii. For the purpose of constructing the risk-free yield curve per currency, an OIS curve (such as Eonia) and a BOR swap curve (such as Euribor 3M) must be considered two different curves. Two BOR curves at different maturities (eg, Euribor 3M and Euribor 6M) must be considered two different curves. An onshore and an offshore currency curve (eg, onshore Indian rupee and offshore Indian rupee) must be
considered two different curves.

(b) The GIRR delta risk factors also include a flat curve of market-implied inflation rates for each currency with term structure not recognised as a risk factor.

i. The sensitivity to the inflation rate from the exposure to implied coupons in an inflation instrument gives rise to a specific capital requirement. All inflation risks for a currency must be aggregated to one number via simple sum.

ii. This risk factor is only relevant for an instrument when a cash flow is functionally dependent on a measure of inflation (eg, the notional amount or an interest payment depending on a consumer price index). GIRR risk factors other than for inflation risk will apply to such an instrument notwithstanding.

iii. Inflation rate risk is considered in addition to the sensitivity to interest rates from the same instrument, which must be allocated, according to the GIRR framework, in the term structure of the relevant risk-free yield curve in the same currency.

(c) The GIRR delta risk factors also include one of two possible cross-currency basis risk factors4 for each currency (ie, each GIRR bucket) with term structure not recognised as a risk factor (ie, both cross-currency basis curves are flat).

i. The two cross-currency basis risk factors are basis of each currency over US dollars or basis of each currency over euros. For instance, an A$-denominated bank trading a ¥/US$ cross-currency basis swap would have a sensitivity to the ¥/US$ basis but not to the ¥//€ basis.

ii. Cross-currency bases that do not relate to either basis over US dollars or basis over euros must be computed either on “basis over US$” or “basis over €” but not both. GIRR risk factors other than for cross-currency basis risk will apply to such an instrument notwithstanding.

iii. Cross-currency basis risk is considered in addition to the sensitivity to interest rates from the same instrument, which must be allocated, according to the GIRR framework, in the term structure of the relevant risk-free yield curve in the same currency.

(d) Vega GIRR: Within each currency, the GIRR vega risk factors are the implied volatilities of options that reference GIRR-sensitive
underlyings; further defined along two dimensions:<sup.5

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, one year, three years, five years, 10 years.

ii. Residual maturity of the underlying of the option at the expiry date of the option: The implied volatility of the option as mapped to two (or one) of the following residual maturity vertices: 0.5 years, one year, three years, five years,10 years.

(e) Curvature GIRR: The GIRR curvature risk factors are defined along only one dimension: the constructed risk-free yield curve (ie, no term structure decomposition) per currency: For example, the Euro, Eonia, Euribor 3M and Euribor 6M curves must be shifted at the same time in order to compute the Euro-relevant risk-free yield curve curvature risk charge. All vertices (as defined for delta GIRR) are to be shifted in parallel. There is no curvature risk charge for inflation and cross-currency basis risks.

(f) The treatment described in paragraph 59(a)(ii) for delta GIRR also applies to vega GIRR and curvature GIRR risk factors.

60. Credit spread risk (CSR) non-securitisation risk factors

(a) Delta CSR non-securitisation: The CSR non-securitisation delta risk factors are defined along two dimensions: the relevant issuer credit spread curves (bond and CDS) and the following vertices: 0.5 years, one year, three years, five years, 10 years to which delta risk factors are assigned.

(b) Vega CSR non-securitisation: The vega risk factors are the implied volatilities of options that reference the relevant credit issuer names as underlyings (bond and CDS); further defined along one dimension:

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, one year, three years, five years, 10 years.

(c) Curvature CSR non-securitisation: The CSR non-securitisation curvature risk factors are defined along one dimension: the relevant issuer credit spread curves (bond and CDS). For instance, the bond-inferred spread curve of Électricité de France and the CDS-inferred spread curve of Électricité de France should be considered a single spread curve. All vertices (as defined for CSR) are to be shifted in parallel.

62. CSR securitisation: non-Correlation Trading Portfolio (“non-CTP”) risk factors

(a) Delta CSR securitisation (non-CTP): the CSR securitisation delta risk factors are defined along two dimensions: tranche, credit spread curves and the following vertices: 0.5 years, one year, three years, five years, 10 years to which delta risk factors are assigned.

(b) Vega CSR securitisation (non-CTP): Vega risk factors are the implied volatilities of options that reference non-CTP credit spreads as underlyings (bond and CDS), further defined along one dimension:

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, 1 year, 3 years, 5 years, 10 years.

(c) Curvature CSR securitisation (non-CTP): the CSR securitisation curvature risk factors are defined along one dimension: the relevant
tranche credit spread curves (bond and CDS). For instance, the bond-inferred spread curve of a given Spanish RMBS tranche and the CDS-inferred spread curve of that given Spanish RMBS tranche would be considered a single spread curve. All the vertices are to be shifted in parallel.

63. CSR securitisation: Correlation Trading Portfolio (“CTP”) risk factors

(a) Delta CSR securitisation (CTP): The CSR correlation trading delta risk factors are defined along two dimensions: the relevant underlying credit spread curves (bond and CDS) and the following vertices: 0.5 years, one year, three years, five years, 10 years to which delta risk factors are assigned.

(b) Vega CSR securitisation (CTP): The vega risk factors are the implied volatilities of options that reference CTP credit spreads as underlyings (bond and CDS), further defined along one dimension:

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, one year, three years, five years, 10 years.

(c) Curvature CSR securitisation (CTP): The CSR correlation trading curvature risk factors are defined along one dimension: the relevant
underlying credit spread curves (bond and CDS). For instance, the bond-inferred spread curve of a given name within an iTraxx series and the CDS-inferred spread curve of that given underlying would be considered a single spread curve. All the vertices are to be shifted in parallel.

64. Equity risk factors

(a) Delta equity: The equity delta risk factors are all the equity spot prices and all the equity repurchase agreement rates (equity
repo rates).

(b) Vega equity: The equity vega risk factors are the implied volatilities of options that reference the equity spot prices as underlyings. There is no vega risk capital charge for equity repo rates. Vega risk factors are further defined along one dimension:

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, one year, three years, five years, 10 years.

(c) Curvature equity: The equity curvature risk factors are all the equity spot prices. There is no curvature risk charge for equity repo rates.

65. Commodity risk factors

(a) Delta commodity: The commodity delta risk factors are all the commodity spot prices depending on contract grade6 of the commodity, legal terms with respect to the delivery location7 of the commodity and time to maturity of the traded instrument at the following vertices: 0 years, 0.25 years, 0.5 years, one year, two years, three years, five years, 10 years, 15 years, 20 years, 30 years.

(b) Vega commodity: The commodity vega risk factors are the implied volatilities of options that reference commodity spot prices as underlyings. No differentiation between commodity spot prices by maturity of the underlying, grade or delivery location is required. The commodity vega risk factors are further defined along one dimension:

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, one year, three years, five years, 10 years.

(c) Curvature commodity: The commodity curvature risk factors are defined along only one dimension: the constructed curve (ie, no term structure decomposition) per commodity spot prices. All vertices (as defined for delta commodity) are to be shifted in parallel.

66. Foreign exchange (FX) risk factors:

(a) Delta FX: All the exchange rates between the currency in which an instrument is denominated and the reporting currency.

(b) Vega FX: For the purpose of vega risk, the foreign exchange risk factors are the implied volatilities of options that reference exchange rates between currency pairs; further defined along one dimension:

i. Maturity of the option: The implied volatility of the option as mapped to one or several of the following maturity vertices: 0.5 years, one year, three years, five years, 10 years.

(c) Curvature FX: All the exchange rates between the currency in which an instrument is denominated and the reporting currency.

(d) No distinction is required between onshore and offshore variants of a currency for all FX delta, vega and curvature risk factors.

Authors’ comment: Calculations of all risk factors under the SBA framework must use the vertices prescribed in FRTB-SA with linear interpolation used to define specified vertices falling between the SBA prescribed levels. This guidance reinforces BCBS’s goal of ensuring the SA framework remains a standardised benchmark across jurisdictions, but will undoubtedly create additional burdens for banks which, for legitimate reasons, set vertices at levels that differ from the SBA prescriptions in the IMA models. The implication for banks employing IMA is that they will not only require two separate model environments, SA and IMA, but may also require two differing data measurement frameworks feeding their SA and IMA RTDs.

Relevant provisions: Paragraph 68 of the January 2016 market risk framework

BCBS response: In the case where options do not have a specified maturity (eg, cancellable swaps), the bank must assign those options to the longest prescribed maturity vertex for vega risk sensitivities and also assign such options to the residual risk add-on.

In the case of the bank viewing the optionality of the cancellable swap as a swaption, the bank must assign the swaption to the longest prescribed maturity vertex for vega risk sensitivities (as it does not have a specified maturity) and derive the residual maturity
of the underlying of the option accordingly.

Referenced FRTB text:

68. Vega risk sensitivities:

(a) The option-level vega risk sensitivity to a given risk factor is the product (ie, multiplication) of the vega and implied volatility of
the option.8 To determine this product, the bank must use the instrument’s vega and implied volatility contained within the pricing models used by the independent risk control unit of a bank.

(b) The portfolio-level vega risk sensitivity to a given vega risk factor is equal to the simple sum of option-level vega risk sensitivities
to that risk factor, across all options in the portfolio.

(c) The following sets out how vega risk sensitivities are to be derived in specific cases:

i. With regard to options that do not have a maturity, assign those options to the longest prescribed maturity vertex, and assign these options also to the residual risks add-on;

ii. With regard to options that do not have a strike or barrier and options that have multiple strikes or barriers, apply the mapping to strikes and maturity used internally to price the option, and assign those instruments also to the residual risks add-on;

iii. With regard to CTP securitisation tranches that do not have an implied volatility, do not compute vega risk for such an instrument. Such instruments may not, however, be exempt from delta and curvature risk charges.

iv. Treatment of index instruments and multi-underlying options.

Authors’ comment: SBA risk capital to be held against cancellable swaps will punitive. Since there is no specified maturity, banks will be required to assign not only vega risk capital at the longest potential vertex but also a residual risk add on. As a result, we expect the use of cancellable swaps in the global bank market to be limited over time.

BCBS response: Both sticky delta and sticky strike approaches are permitted.
Referenced FRTB text:
71. When computing a first-order sensitivity for instruments subject to optionality, banks should assume that the implied volatility remains constant, consistent with a “sticky delta” approach. This concept is illustrated in Figure 14.1.

Authors’ comment: Sticky strike assumes that the volatility skew for an option remains constant across the strike values. Sticky delta assumes that the volatility skew remains unchanged with the option’s moneyness – ie, the relative position of a current or future price relative to the strike price. Both sticky price and delta implicitly assume that volatility remains range-bound.This may not always be true in practice, but the otherwise conservative SA

framework provides for adequate capital charge cushion despite the assumptions about underlying volatilities.

Relevant provisions: paragraph 132 of the January 2016 market risk framework.

BCBS response: The risk weight for the curvature risk charge should be the highest prescribed delta risk weight for each of the delta risk factors that are shocked together to determine the curvature risk charge.

Referenced FRTB text:
132. For GIRR, CSR and commodity curvature risk factors, the curvature risk weight is the parallel shift of all the vertices for each curve
based on the highest prescribed delta risk weight for each risk class. For example, in the case of GIRR the risk weight assigned to the 0.25-
year vertex (ie, most punitive vertex risk weight) is applied to all the vertices simultaneously for each risk-free yield curve (consistent
with a “translation”, or “parallel shift”, risk calculation).

Authors’ comment: This question relates to how parallel shifts of delta risks are conducted to derive the curvature risk. The question was whether delta risk weights should be derived from each risk factor and then shocked together, or whether the delta risk weights should be the highest prescribed risk weights for each delta risk factor. BCBS specifies that the former methodology should be used.