FRTB – Fundamental Review of the Trading Book Sensitivities-based Method of the Standardised Approach: the step-by-step recipe

Françoise Caclin Founder, Fimarkets 1 FRTB – Standardised Approach Step by Step

Our previous article outlined the as “the document”. The purpose of reform of monitoring, this presentation is obviously not defined by the Basel Committee to reproduce this document, but to and known as FRTB. In this article, propose an approach focused on the we will review the standardised risk major concepts, and hopefully, more calculation method based on risk educational. sensitivities that will come into effect for from 2020 onwards. It Overall, the calculation of market should be remembered that even risk using the standardised method banks, which, with the agreement of consists in determining a capital their supervisory authority, opt for charge per risk class and aggregating an internal method, will have to be them to determine the overall capital able to calculate their and charge for market risk. To this are capital charge using the standardised added the charge for the risk of method. This is one reason why it default, as well as the additional makes sense to take an interest in it, charge for the residual risk. the other being that this approach in This article will focus in particular on itself is a good basis for a pedagogical the calculation of the market risk. We view of market risk. could draw a parallel with a cooking The standard method is presented recipe: the important thing is to have in considerable detail in the January the right ingredients... and then stick 2016 Basel Committee document, to the proper preparation steps. “Minimum Capital requirements for market risk”, hereinafter referred to FRTB – Standardised Approach Step by Step 2

The Ingredients

It is hard to choose an of because the definition of risk correlations...) - and vice versa! presentation of the different indicators (Delta, Vega and Curvature) calculation parameters (ingredients), refers to risk factors (classes, factors,

Risk class Computation parameters Risk factors Interest-rate Risk factors Delta Credit spread risk: non securitisation Buckets Credit spread risk: (CTP) Weights Vega Credit spread risk: securitization (non-CTP) Intra-bucket correlation factors Equity Inter-bucket correlation factors Curvature Commodity Foreign exchange

Risk classes

7 classes of market risk are defined: A special effort was made to analyse the credit spread risk, with a • General (GIRR) detailed classification of the various • Credit Spread Risk (CSR), which is subdivided into three categories: instruments that gave rise to the 2008 financial crisis. - Risk non-related to securitisation - Risk related to securitisation within the Correlation Trading Portfolio (CTP) - Risk related to securitisation outside the Correlation Trading Portfolio • • •

Risk factors

For each risk class and indicator Examples of risk factors for the relation to the ’s accounting (Delta, Vega and Curvature, see Delta calculation . below), the document identifies the risk factors that must be taken • Interest rate risk: risk factors are Risk factors for the calculation of into account for the calculation of determined by rate curves, and Vega and Curvature sensitivities, as well as the weights to identified for a predefined set of • Vega: the risk factors are the be applied to calculate the “weighted points (3M, 6M, 1Y, 2Y, 3Y, etc.) or implied volatilities of the options sensitivities” whose aggregation will vertices. having an underlying Delta risk directly result in the capital charge. • : credit spread curves (of factor (interest rate options for Generally speaking, a is bonds or CDS) for a predefined set interest rate risk, equity options, an observable or measurable market of points etc.). data that is likely to influence the • Equities: share prices and repo • Curvature: risk factors are and therefore the profit rates modelled on the risk factors used in or loss generated by a financial to calculate the Delta. instrument. • Commodities: commodity prices according to the different The risk factors for the different maturities of the contracts risk classes and the 3 indicators are negotiated (spot, 3M, 6M, etc.) detailed in sections 59 to 66 of the document. • Foreign exchange: of the currency in which the portfolio instruments are traded in 3 FRTB – Standardised Approach Step by Step

Buckets and weights

The document defines a bucket as a yield & non-rated) on the one hand • Commodities: buckets correspond, set of instruments of the same risk and by economic sector (sovereign, quite logically, to categories of class sharing the same characteristics , etc.) on the other hand. raw materials (energy, metals, and therefore the same “risk profile”. agriculture...) For example, in the case of the • Exchange rate: currency pairs interest rate risk, buckets in the sense The buckets are the same for the • Credit (non-CTP securitisation) of FRTB are simply . This 3 risk indicators (Delta, Vega and : buckets also correspond to a does not correspond to the usual Curvature). However, each bucket two-level classification, with credit meaning in the trading room, but in has its own risk weights for each quality at the first level and a this context, buckets correspond to indicator. securitisation “sector” at the second time intervals. level: RMBS, CMBS, ABS, CLO... The product of the sensitivity • Interest rates: buckets correspond calculated for a given risk factor and • Equities: buckets correspond to the different currencies of the weighting corresponding to the to a 3-level classification of the risk bucket, or weighted sensitivity, • Credit (non-related to assets: market cap (large / small), gives a capital charge that could securitisation and securitisation economy (emerging, advanced) be called “elementary” (for a risk within the CTP): buckets and finally economic sector (retail indicator, a risk factor, a risk bucket). correspond to a two-level goods services, telecom and classification of credit risk, by industry...). quality ( grade, high-

Correlations parameters

To compute the overall capital two levels of aggregation. sensitivities across buckets in order charge for a class and a risk indicator, to obtain the overall sensitivity for the weighted sensitivities must be • “Intra-bucket” correlation the risk class. These coefficients are aggregated. This is not done by parameters are used to aggregate designated by the Greek letter. simply summing them up, as both sensitivities within the same portfolio diversification and the risk bucket in a first step. These The definition of correlation propensity of risk factors to fluctuate parameters are designated by the parameters varies in complexity.For simultaneously must be taken into Greek letter in the documentation. example, for credit spread risk, the account. This is where correlation correlation parameter Pkl between 2 • “Inter-bucket” correlation parameters come into play. There are sensitivities WSK and WSl within the parameters are used to aggregate two categories corresponding to the same bucket is defined as follows:

Where:

(name) • Pkl = 1 if the two issuers of k and l are identical, 35% otherwise P = P (name) . P (tenor) . P (basis) kl kl kl kl (tenor) • Pkl = 1 if the two vertices of the credit curve are identical for k and l, 65% otherwise

(basis) • Pkl = 1 if the 2 sensitivities are related to the same curve, 99.9% otherwise

For example (as presented in the Buckets, weights and correlation • In sections 122 to 128 for the FRTB document), the correlation parameters are described calculation of the Vega parameter between the sensitivity on the Apple 5Y curve and the • In sections 73 to 121 for the • In sections 131 to 133 for the sensitivity on the CDS Google 10Y calculation of the Delta calculation of the Curvature curve is 35%. 65%. 99.9% = 22.73%. FRTB – Standardised Approach Step by Step 4

The formulas that use correlation parameters to calculate aggregate sensitivities are described below. Risk indicators

For each risk class, three aggregated indicators or “sensitivities” must be calculated: Delta, Vega and Curvature, the last two applying only to options and instruments with embedded optionality.

Delta The Delta corresponds to the point) will be calculated. For example, for an interest-rate sensitivity of the value of a position to sensitive instrument i (e. g. a bond), a variation of one basis point (0.01%) The formulas for calculating the the Delta (PV01) will be calculated as in the risk factor analysed. In the case Delta for the different risk classes follows: of the interest rate risk, for example, are detailed in section 67 of the a “PV01” (Price Value of one basis document.

Where:

• rt is the risk-free yield curve at point t V (r + 0.00001, cs ) - V (r ,cs ) i t t i t t • cst is the credit spread curve for the instrument S k,rt = considered at vertex t 0.0001 • Vi is the function that calculates the market value of the instrument i based on the risk-free interest rate and the credit spread

Note that the sensitivity to the credit spread, the CS01, will be calculated in a similar way but by varying the spread instead of the risk- free interest rate:

Note: in the following calculations, sensitivities are multiplied by the weights, resulting in the so-called “weighted” sensitivities. Vi (rt, cst + 0.00001) - Vi(rt,cst) These weights are expressed as a percentage, hence the division by S k,rt = 0.0001 at this stage... 0.0001

Vega The Vega is equal to the product of In times of market stress, volatility also increase, which again impacts the vega and the implied volatility increases significantly for most the value of the assets... That is of the ; knowing that the vega asset classes. Consequently, market why regulators have decided to itself represents the sensitivity of the participants buy options in order to specifically address the volatility of option price to the implied volatility. their portfolios, so that the assets in the monitoring of market price of options and their . 5 FRTB – Standardised Approach Step by Step

Curvature Finally, the Curvature risk consists significant variation of the market The capital charge for curvature risk of applying a significant change (a value of the instrument, deduction for risk factor k is calculated as follows: “shock”) upwards and downwards to made of the Delta. each risk factor, and retaining the most

(curvature) (RW + ( ( curvature ) Vi xk - vi ( xk ) - RWk . sik i{ ( ( { CVR k = - min RW(curvature) - ( curvature ) Vi xk ( ( - vi ( xk ) - RWk . sik i{ ( ( {

Where:

• i is an instrument subject to curvature risk associated with risk factor k

• xk is the current value of the risk factor k

• Vi ( xk ) is the price of instrument i for the current value of the risk k

• (curvature) and (curvature) denote the price of (RW - ( (RW + ( instrument i after the risk Vi xk Vi xk factor k has been shifted ( ( ( ( upward and downward respectively.

( curvature ) • RWk is the weight assigned to the risk factor k as defined by documentation

• Sik is the delta of instrument i for risk factor k

While the delta reflects the sensitivity to small variations in risk factors, the curvature seeks to capture the effect of a large variation (a “shock”) of this same risk factor. FRTB – Standardised Approach Step by Step 6

Selection of calculation parameters

Finally, it should be noted that indicator determines a different set the selection of the calculation of risk factors, buckets, weights and parameters is multidimensional: each correlation parameters. pair defined by a risk class and an

Preparation

Overall, the calculation steps are as follows:

• Identify risk factors

• Calculate weighted sensitivities

• Aggregate sensitivities

Calculate the weighted Aggregate sensitivities by Identify the risk factors for each sensitivities (Delta, Vega, bucket then between buckets position Curvature) of each position to with 3 correlation scenarios and the different risk factors retain the maximum result 7 FRTB – Standardised Approach Step by Step

Identify risk factors Calculate weighted Aggregate sensitivities The first step is to identify, for all sensitivities by risk class instruments held in the trading The aggregation by risk class is done portfolio, the risk factors that apply Each net sensitivity is then assigned in a similar way using the results of and which risk buckets they belong the risk weight provided in the the previous step and the correlation to. documentation for the relevant risk factor + risk bucket to give the parameters between buckets within Examples: weighted sensitivity: the same risk class: • A in US dollar with

a residual maturity of 1 year, held WS = RW S 2 k k k Delta = K + y S S in a portfolio accounted for in Euro, k bc b c is sensitive to interest rate risk on b b c=b the 3M, 6M and 1Y maturities, but Aggregate sensitivities by also to credit spread risk on the Where: issuer and also to the EUR / USD risk class

exchange rate risk. • Sb (Sc) = k WSk for all risk factors within bucket b (c) • A share traded in British Pound is Aggregate sensitivities

sensitive to the equity risk on the by bucket • ybc is the correlation issuer’s category and also to the parameter between EUR / GBP exchange rate risk. Weighted sensitivities by risk bucket buckets b and c must then be aggregated. For Delta • An ABS (Asset-Backed ) is and Vega, the formula for calculating sensitive to credit spread risk for the capital charge for bucket b is as Remarks the underlying debt category. follows: The aggregation formulas for Delta and Vega are the same; they

2 differ slightly for the curvature. Calculate net sensitivities by K b = WS k + pklWSkWSl risk factor The documentation also provides k k k=l alternative scenarios for the case where the quantity under the square The next step is to calculate net root is negative. sensitivities by risk factor for the Where: instruments in the portfolio. The Scenarios term “net” is important, meaning that • k and l representrisk factors we calculate the arithmetic sum of This dual aggregation process

all Delta (respectively all Vega and • Pkl is the correlation must be carried out 3 times per Curvatures) calculated on a given parameter between risk risk class and indicator, with three risk factor in the trading portfolio. factors k and l different correlation scenarios: As a result, sensitivities in opposite low, medium and high. For each • WS and WS are weighted directions for a given risk factor k l indicator, we will retain the correlation sensitivities to k et l risk compensate each other, which makes scenario with the highest end result factors sense from the risk point of view: (Delta+Vega+Curvature, for all risk a position sensitive to a risk factor classes). in one direction can be hedged by another position that varies in the Using 3 scenarios makes it possible opposite direction. In other words, to take into account the fact that in within the same risk class and for times of market stress, correlations the same indicator, the portfolio’s between risk factors may increase diversification effect is fully exploited or decrease. The “medium” scenario to achieve capital charge “savings” corresponds to the “basic” values of when exposed to market risk. the correlation parameters defined in the literature. The “low” scenario consists of multiplying all these coefficients by 0.75, and the “high” scenario of multiplying them all by 1.25. FRTB – Standardised Approach Step by Step 8

Calculate the final capital charge

The Delta (respectively Vega and any impact of portfolio diversification since there are three correlation Curvature) is simply equal to the sum on the capital charge. The total scenarios, and retain the highest one of the Delta (respectively Vega and capital charge is simply equal to the as the capital charge for market risk. Curvature) by risk class. At this stage sum Delta + Vega + Curvature. We of the calculation, there is no longer will calculate three capital charges

Delta risk factors, risk buckets Vega risk factors, risk buckets Curvature risk factors, risk and risk weights and risk weights buckets and risk weights

Delta weighted Vega weighted Curvature weighted sensitivities sensitivities sensitivities

Delta Correlation matrices Vega Correlation matrices Curvature Correlation matrices (High, Medium, Kow) (High, Medium, Kow) (High, Medium, Kow)

Market risk charge = Max ( Delta (High) + Vega (High) + Curvature (High), Delta (Medium) + Vega (Medium) + Curvature (Medium), Delta (Low) + Vega (Low) + Curvature (Low))

In addition to this capital charge for • The default risk charge market risk, which stems directly from sensitivities to risk factors, it will • The residual risk add-on for residual be necessary to add: risk 9 FRTB – Standardised Approach Step by Step

…in order to obtain the final capital charge.

GIRR General Interest Rate Risk

CSR Credit Spread Risk non securitisation

CSR Credit Spread Risk securitisation

CSR Credit Spread Risk securitisations (non correlation + MarketRisk Charge trading portfolio)

Default Risk + Final MarketRisk Charge EquityRisk ResidualRiska dd-on

CommodityRisk

Foreign Exchange risk Global Footprint

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