11 Sensitivities-based Method (SBM)
11.1 Main Definitions
69 Risk class means one of the following seven risk classes for the sensitivities-based method: (i) general interest rate risk, (ii) credit spread risk for non-securitisation, (iii) credit spread risk for securitisations13 (non-correlation trading portfolio), (iv) credit spread risk for securitisations (correlation trading portfolio), (v) equity risk, (vi) commodity risk and (vii) foreign exchange risk.
70 Risk factor means a variable, e.g. a tenor of an interest rate curve or an equity price, that affects the value of an instrument falling within the scope of the risk factor definitions in subsection 12. Risk factors are mapped to a risk class.
71 Risk position is the main input that enters the risk charge computation. For delta and vega risks, it is a sensitivity to a risk factor. For curvature risk, it is based on the worse of upward and downward stress scenarios.
72 Bucket means a set of risk factors within one risk class which are grouped together by common characteristics, as defined in paragraphs 141, 148, 154, 157, 163 and 174.
73 Risk charge is the amount of capital that an AI should hold as a consequence of the risks it takes; it is computed as an aggregation of risk positions first at the bucket level, and then across buckets within a risk class defined for the sensitivities-based method.
11.2 Components of the SBM
74 An AI should calculate the risk charge for market risk under the sensitivities-based method by aggregating the following risk measures:
Delta risk which captures the risk of changes in the market value of an AI’s position due to movements in its non-volatility linear risk factors, as defined in paragraphs 85, 95, 99, 103, 106, 109, 112 and 113;
Vega risk which captures the risk of changes in the market value of an AI’s position due to movements in its volatility linear risk factors, as defined in paragraphs 92, 96, 100, 104, 107, 110 and 115; and
13 The term “securitisation” has the same meaning of “securitization exposure” as defined in section 2(1) of the BCR.
Curvature risk which captures the risk of changes in the market value of an AI’s position due to movements in its non-volatility risk factors not captured by the delta risk, as defined in paragraphs 93, 97, 101, 105, 108, 111, 116 and 117.
Curvature risk is based on two stress scenarios involving an upward and a downward shock to a given risk factor.
75 The above three risk measures specify risk weights to be applied to the regulatory risk factor sensitivities. To calculate the overall capital charge, the risk-weighted sensitivities are aggregated using specified correlation parameters to recognise diversification benefits between risk factors. In order to address the risk that correlations may increase or decrease in periods of financial stress, three risk charge figures should be calculated for each risk class defined under the sensitivities-based method (see paragraphs 83 to 84 for details), based on three different scenarios on the specified values for the correlation parameter 𝜌𝑘𝑙 (i.e. correlation between risk factors within a bucket) and 𝛾𝑏𝑐 (i.e. correlation between risk factors across buckets within a risk class). There should be no diversification benefit recognised between individual risk classes.
11.3 Instrument Prices and Sensitivity Calculation
76 In calculating the capital charge under the sensitivities-based method, an AI should determine each delta and vega sensitivity and curvature scenario based on instrument prices or pricing models that an independent risk control unit within the AI uses to report market risks or actual profits and losses to senior management.
77 A key assumption of the Standardised Approach for market risk is that an AI’s pricing model used in actual profit and loss reporting provides an appropriate basis for the determination of regulatory capital charges for all market risks. To ensure such adequacy, an AI should at minimum fulfil the requirements in section 4A of the BCR and the Supervisory Policy Manual (SPM) Module CA-S-10 “Financial Instrument Fair Value Practices”.
11.4 Instruments Subject to Delta, Vega and Curvature
78 All instruments held in trading desks and subject to the sensitivities-based method (i.e. excluding instruments where the value at any point in time is purely driven by an exotic underlying as set out in subsection 16), are subject to delta risk capital charge.
Additionally:
An instrument with optionality, or with non-zero vega sensitivities14 is subject to risk charges for vega risk and curvature risk;
An instrument with an embedded prepayment option15 is an instrument with optionality. Accordingly, the embedded option is subject to risk charges for vega and curvature risk with respect to the interest rate risk and credit spread risk (non-securitisation and securitisation) risk classes. When the prepayment option is a behavioural option, the instrument may also be subject to the residual risk add-on as per paragraph 198. The pricing model of the AI should reflect such behavioural patterns where relevant. Instruments in the securitised portfolio may have embedded prepayment options as well. In this case they may be subject to the residual risk add-on;
Instruments whose cash flows cannot be written as a linear function of underlying notional are subject to vega risk and curvature risk charges. For example, the cash flows generated by a plain-vanilla option cannot be written as a linear function (as they are the maximum of the spot and the strike).
Therefore all options are subject to vega risk and curvature risk. Instruments whose cash flows generated by a coupon-bearing bond can be written as a linear function, are not subject to vega risk nor curvature risk charges.
Curvature risks may be calculated for all instruments subject to delta risk, not limited to those subject to vega risk as specified above. For example, where an AI manages the non-linear risk of instruments with optionality and other instruments holistically, the AI may choose to include instruments without optionality in the calculation of curvature risk. This treatment is allowed subject to the following restrictions: (i) use of this approach shall be applied consistently through time; and (ii) curvature risk should be calculated for all instruments subject to the sensitivities-based method.
11.5 Delta and Vega Risks
79 An AI should apply the delta and vega risk factors defined in subsection 12 to calculate the risk charge for delta and vega risks.
80 For each risk class, an AI should determine its instruments’ sensitivity to a set of prescribed risk factors, risk-weight those sensitivities, and aggregate the resulting
14 There are some instruments that are not options, but are still subject to risk charges for vega risk and curvature risk. For example, convexity adjustments on constant maturity swaps (CMS) can generate significant vega risk, which are subject to vega and curvature risk charges.
15 An instrument with a prepayment option is a debt instrument which grants the debtor the right to repay part or the entire principal amount before the contractual maturity without having to compensate for any foregone interest. The debtor can exercise this option with a financial gain by obtaining funding over the remaining maturity of the instrument at a lower rate in other ways in the market.
risk-weighted sensitivities separately for delta and vega risk using the following step-by-step approach.
Step 1: For each risk factor, a sensitivity is determined as set out in subsection 12.
Step 2: Sensitivities to the same risk factor should be netted to give a net sensitivity 𝑠𝑘 across all instruments in the portfolio to each risk factor k. In calculating the net sensitivity, all sensitivities to the same given risk factor (e.g. all sensitivities to the 1-year tenor point of the HKD 3-month swap curve) from instruments of opposite direction should offset, irrespective of the instrument from which they derive.
Step 3: The risk-weighted sensitivity 𝑊𝑆𝑘 is the product of the net sensitivity 𝑠𝑘 and the corresponding risk weight 𝑅𝑊𝑘 as defined in subsections 13 and 14.
𝑊𝑆𝑘= 𝑅𝑊𝑘𝑠𝑘
Step 4: Within bucket aggregation: The risk position for delta (respectively vega) bucket 𝑏, 𝐾𝑏, should be determined by aggregating the weighted sensitivities to risk factors within the same bucket using the corresponding prescribed correlation 𝜌𝑘𝑙 set out in the following formula: determined from risk positions aggregated between the delta (respectively vega) buckets within each risk class, using the corresponding prescribed correlations 𝛾𝑏𝑐 as set out in the following formula: risk capital charge using an alternative specification whereby Sb = max [min (∑ WSk k,Kb), − Kb] for all risk factors in bucket b and Sc = max [min (∑ WSk k,Kc), − Kc] for all risk factors in bucket c.
11.6 Curvature Risk
81 An AI should apply two stress scenarios on given risk factors which are defined in subsection 12 to calculate the risk charge for curvature risk. The two stress scenarios are to be computed per risk factor (an upward and a downward shock) with the delta effect being removed. They are shocked by risk weights and the worst loss is aggregated by correlations provided in subsection 15.
82 An AI should apply the following step-by-step approach to each risk class separately to capture curvature risk:
Step 1: For each instrument sensitive to curvature risk factor k, an upward shock and a downward shock should be applied to k. For example for GIRR, all tenors of all the curves within a given currency (e.g. HKD 1-month swap curve, HKD 3-month swap curve) should be shifted upward. The potential loss, after deduction of the delta risk positions, is the outcome of the upward scenario (𝐶𝑉𝑅𝑘+). The same approach should be followed on a downward scenario (𝐶𝑉𝑅𝑘−). If the price of an option depends on several risk factors, the curvature risk is determined separately for each risk factor.
Step 2: The net curvature risk capital charge, determined by the values of 𝐶𝑉𝑅𝑘+ and 𝐶𝑉𝑅𝑘− for risk factor k is calculated as follows.
𝐶𝑉𝑅𝑘+ = − ∑ {𝑉𝑖(𝑥𝑘𝑅𝑊(𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒)+) − 𝑉(𝑥𝑘) − 𝑅𝑊𝑘𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒∙ 𝑠𝑖𝑘}
𝑖
𝐶𝑉𝑅𝑘− = − ∑ {𝑉𝑖(𝑥𝑘𝑅𝑊(𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒)−) − 𝑉(𝑥𝑘) + 𝑅𝑊𝑘𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒∙ 𝑠𝑖𝑘}
𝑖
This calculates the aggregate incremental loss beyond the delta capital charge for the prescribed shocks, 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 (i.e. “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 191; 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 general interest rate risk (GIRR), credit spread risk (CSR) and commodity risk classes:
- 𝑅𝑊𝑘(𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒)
is the risk weight for curvature risk factor k for instrument i determined in accordance with paragraph 193; and
- 𝑠𝑖𝑘 is the sum of delta risk sensitivities to all tenors of the relevant curve(s) of instrument i with respect to curvature risk factor k.
Step 3: Within bucket aggregation: The curvature risk exposure should be aggregated within each bucket using the corresponding prescribed correlation 𝜌𝑘𝑙 as set out in the following formula:
the bucket level capital charge (𝐾𝑏) is determined as the greater of the capital charge under the upward scenario (𝐾𝑏+) and the capital charge under the downward scenario (𝐾𝑏−). Notably, the selection of upward and downward scenarios is not necessarily the same across the high, medium and low correlations scenarios specified in paragraph 83.
- Where 𝐾𝑏=𝐾𝑏+, this shall be termed “selecting the upward scenario”. aggregated across buckets within each risk class, using the corresponding prescribed correlation 𝛾𝑏𝑐.
𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 𝑟𝑖𝑠𝑘 = √𝑚𝑎𝑥(0, ∑ 𝐾𝑏2+
𝑏
∑ ∑ 𝛾𝑏𝑐
𝑐≠𝑏
𝑆𝑏𝑆𝑐
𝑏
𝜓(𝑆𝑏, 𝑆𝑐) )
where:
𝑆𝑏= ∑ 𝐶𝑉𝑅𝑘 𝑘+ for all risk factors in bucket b, when the upward scenario has been selected for bucket b above. 𝑆𝑏 = ∑ 𝐶𝑉𝑅𝑘 𝑘− otherwise ; and
𝜓(𝑆𝑏, 𝑆𝑐) takes the value 0 if 𝑆𝑏 and 𝑆𝑐 both have negative signs. In all other cases, 𝜓(𝑆𝑏, 𝑆𝑐) takes the value of 1.
11.7 Correlation Scenarios and Aggregation of Risk Charges
83 In order to address the risk that correlations increase or decrease in periods of financial stress, an AI will be required to calculate three risk charge figures for each risk class, corresponding to three different scenarios on the specified values for the correlation parameters 𝜌𝑘𝑙 (correlation between risk factors within a bucket) and 𝛾𝑏𝑐 (correlation across buckets within a risk class).
the “high correlations” scenario, whereby the correlation parameters 𝜌𝑘𝑙 and 𝛾𝑏𝑐 that are specified in subsections 13, 14 and 15 are uniformly multiplied by 1.25, with 𝜌𝑘𝑙 and 𝛾𝑏𝑐 subject to a cap at 100%;
the “medium correlations” scenario, whereby the correlation parameters 𝜌𝑘𝑙 and 𝛾𝑏𝑐 remain unchanged from those specified in subsections 13, 14 and 15;
and
the “low correlations” scenario whereby the corresponding prescribed correlations are the correlations given in subsections 13, 14 and 15 are replaced by 𝜌𝑘𝑙𝑙𝑜𝑤= max(2 ∙ 𝜌𝑘𝑙 ‒ 100%, 75% ∙ 𝜌𝑘𝑙) and 𝛾𝑏𝑐𝑙𝑜𝑤 = max(2 ∙ 𝛾𝑏𝑐 ‒ 100%, 75% ∙ 𝛾𝑏𝑐).
84 The total capital charge under the sensitivities-based method is aggregated as follows:
For each scenario, an AI should simple sum up the delta, vega and curvature risk charges for all risk classes to determine the overall risk charge for that scenario.
The ultimate risk capital charge is the largest capital charge from the three scenarios.
- For the calculation of capital charge for all instruments in all trading desks using the Standardised Approach as set out in paragraph 15, the capital charge is calculated for all instruments in all trading desks.
- For the calculation of capital charge for each trading desk using the Standardised Approach as if that desk were a standalone regulatory portfolio as set out in paragraph 15, the capital charges under each correlation scenario are calculated and compared at each trading desk level, and the maximum for each trading desk is taken as the capital charge.