## Introduction

Our previous article
outlined the reform of **market risk monitoring**, defined by the Basel Committee and known as FRTB. In this
article we will review the standardised risk calculation method based on **risk
sensitivities **that will come into effect for banks from 2020 onwards. It
should be remembered that even banks which, with the agreement of their
supervisory authority, opt for an internal method will have to be able to
calculate their market risk and capital charge using the standardised method.
This is one reason why it makes sense to take an interest in it, the other being
that this approach is in itself a good basis for a pedagogical view of market
risk.

The standard method is presented in considerable detail in the January 2016 Basel Committee document "Minimum Capital requirements for market risk", hereinafter referred to as "the document". The purpose of this presentation is obviously not to reproduce this document, but to propose an approach focused on the major concepts, and hopefully, more educational.

Overall, the calculation of market risk using the standardised method consists in determining a capital charge per risk class and aggregating them to determine the overall capital charge for market risk. To this are added the charge for the risk of default, as well as the additional charge for the residual risk.

In this article we will focus in particular on the
calculation of the **market risk**. We could draw a parallel with a cooking
recipe: the important thing is to have the right ingredients... and then stick
to the preparation steps.

## The ingredients

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

Risk class | Computation parameters | Risk factors |

Interest-rate |
Risk factors Buckets Weights |
Delta |

Credit spread risk: non securitisation | ||

Credit spread risk: securitization (CTP) | Vega | |

Credit spread risk: securitization (non-CTP) |
Intra-bucket correlation factors Inter-bucket correlation factors | |

Equity | Curvature | |

Commodity | ||

Foreign exchange |

### Risk classes

7 classes of market risk are defined:

- General
**Interest Rate**Risk (GIRR) **Credit Spread Risk**(CSR), which is subdivided into three categories:- 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
**Equity**risk**Commodity**risk**Foreign Exchange**risk

A special effort was made to analyse the credit spread risk, with a detailed classification of the various instruments that gave rise to the 2008 financial crisis.

### Risk factors

For each risk class and indicator (Delta, Vega and
Curvature, see below), the document identifies the **risk factors** that must
be taken into account for the calculation of sensitivities, as well as the **
weights** to be applied to calculate the "weighted sensitivities" whose
aggregation will directly result in the **capital charge**.

Generally speaking, a risk factor is an observable or measurable market data that is likely to influence the valuation and therefore the profit or loss generated by a financial instrument.

#### Examples of risk factors for the Delta calculation

**Interest rate risk**: risk factors are determined by rate curves, and identified for a predefined set of points (3M, 6M, 1Y, 2Y, 3Y, etc.) or*vertices*.**Credit risk**: credit spread curves (of bonds or CDS) for a predefined set of points**Equities**: share prices and repo rates**Commodities**: commodity prices according to the different maturities of the contracts negotiated (spot, 3M, 6M, etc.)**Foreign exchange**: exchange rate of the currency in which the portfolio instruments are traded in relation to the bank's accounting currency.

#### Risk factors for the calculation of Vega and Curvature

**Vega**: the risk factors are the implied volatilities of the options having an underlying Delta risk factor (interest rate options for interest rate risk, equity options, etc.).**Curvature**: risk factors are modelled on the risk factors used in to calculate the Delta.

The risk factors for the different risk classes and the 3 indicators are detailed in sections 59 to 66 of the document.

### Buckets and weights

The document defines a **bucket** as a set of
instruments of the same risk class sharing the same characteristics and
therefore the same "risk profile". This term "bucket" can be confusing. For
example, in the case of the interest rate risk, buckets in the sense of FRTB are
simply... currencies. This does not correspond to the usual meaning in the
trading room, where buckets in this context correspond rather to time intervals.

**Interest rates**: buckets correspond to the different currencies**Credit**(non-related to securitisation and securitisation within the CTP): buckets correspond to a two-level classification of credit risk, by quality (investment grade, high-yield & non-rated) on the one hand and by economic sector (sovereign, finance, etc.) on the other hand.**Credit (non-CTP securitisation)**: buckets also correspond to a two-level classification, with credit quality at the first level and a securitisation "sector" at the second level: RMBS, CMBS, ABS, CLO...**Equities**: buckets correspond to a 3-level classification of the assets: market cap (large / small), economy (emerging, advanced) and finally economic sector (retail goods services, telecom and industry...).**Commodities**: buckets correspond, quite logically, to categories of raw materials (energy, metals, agriculture...)**Exchange rate**: currency pairs

The buckets are the same for the 3 risk indicators (Delta,
Vega and Curvature). However, each bucket has its own **risk weights** for
each indicator.

The product of the sensitivity calculated for a given risk
factor and of the weighting corresponding to the risk bucket, or **weighted
sensitivity**, gives a **capital charge** that could be called "elementary"
(for a risk indicator, a risk factor, a risk bucket).

### Correlations parameters

To compute the overall capital charge for a class and a risk indicator, the weighted sensitivities must be aggregated. This is not done by simply summing them up, as both portfolio diversification and the propensity of risk factors to fluctuate simultaneously must be taken into account. This is where correlation parameters come into play. There are two categories corresponding to the two levels of aggregation.

- "Intra-bucket" correlation parameters are used to aggregate sensitivities within the same risk bucket in a first step. These parameters are designated by the Greek letter `rho` in the documentation.
- "Inter-bucket" correlation parameters are used to aggregate sensitivities across buckets in order to obtain the overall sensitivity for the risk class. These coefficients are designated by the Greek letter`gamma`.

The definition of correlation parameters varies in
complexity. For example, for credit spread risk, the correlation parameter ρ_{kl}
between 2 sensitivities `WS_k` and `WS_l`
within the same bucket is defined as follows:

Where:

- `rho_{kl}^{(name)}` = 1 if the two issuers of `k` and `l` are identical, 35% otherwise
- `rho_{kl}^{(t e n o r)}` = 1 if the two vertices of the credit curve are identical for `k` and `l`, 65% otherwise
- `rho_{kl}^{(basis)}` = 1 if the 2 sensitivities are related to the same curve, 99.9% otherwise

For example (as presented in the FRTB document), the correlation parameter between the sensitivity on the Apple 5Y bond curve and the sensitivity on the CDS Google 10Y curve is 35% · 65% · 99.9% = 22.73%.

Buckets, weights and correlation parameters are described:

- In sections 73 to 121 for the calculation of the Delta
- In sections 122 to 128 for the calculation of the Vega
- In sections 131 to 133 for the calculation of the Curvature

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 **sensitivity** of the
value of a position to a variation of one basis point (0.01%) in the risk factor
analysed. In the case of the interest rate risk, for example, a "PV01" (Price
Value of one basis point) will be calculated.

The formulas for calculating the Delta for the different risk classes are detailed in section 67 of the document.

For example, for an interest-rate sensitive instrument `i` (e.g. a bond), the Delta (PV01) will be calculated as follows:

Where:

- `r_t` is the risk-free yield curve at point `t`
- `cs_t` is the credit spread curve for the instrument considered at vertex `t`
- `V_i` 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. These weights are expressed as a percentage, hence the division by 0.0001 at this stage...

#### Vega

The Vega is equal to the product of the vega and the implied volatility of the option; knowing that the vega itself represents the sensitivity of the option price to the implied volatility.

In times of market stress, volatility increases significantly for most asset classes. Consequently, market participants buy options in order to hedge their portfolios, so that the price of options and their volatility also increase, which again impacts the value of the assets... That is why regulators have decided to specifically address the volatility of assets in the monitoring of market risk.

#### Curvature

Finally, the Curvature risk consists of applying a significant change (a "shock") upwards and downwards to each risk factor, and retaining the most significant variation of the market value of the instrument, deduction made of the Delta.

The capital charge for curvature risk for risk factor k is calculated as follows:

Where:

- `i` is an instrument subject to curvature risk associated with risk factor `k`
- `x_k` is the current value of the risk factor `k`
- `V_k(x_k)` is the price of instrument `i` for the current value of the risk `k`
- `V_i(x_k^{(RW^{(curvature)-})})` and `V_i(x_k^{(RW^{(curvature)+})})` denote the price of instrument i after the risk factor k has been shifted upward and downward respectively.
- `RW_k^{(curvature)}` is the weight assigned to the risk factor k as defined by documentation

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.

### Selection of calculation parameters

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

## Preparation

Overall, the calculation steps are as follows:

- Identify risk factors
- Calculate weighted sensitivities
- Aggregate sensitivities

### Identify risk factors

The first step is to identify, for all instruments held in
the trading portfolio, the **risk factors** that apply and which risk **
buckets** they belong to.

Examples :

- A corporate bond in US dollar with a residual maturity of 1 year, held in a portfolio accounted for in Euro, is sensitive to interest rate risk on the 3M, 6M and 1Y maturities, but also to credit spread risk on the issuer and also to the EUR / USD exchange rate risk.
- A share traded in British Pound is sensitive to the equity risk on the issuer's category and also to the EUR / GBP exchange rate risk.
- A swaption is sensitive to interest rate risk but also to the volatility of market rates over the option's life.
- An ABS (Asset-Backed Security) is sensitive to credit spread risk for the underlying debt category.

### Calculate net sensitivities by risk factor

The next step is to calculate **net sensitivities by risk
factor** for the instruments in the portfolio. The term "net" is important,
meaning that we calculate the arithmetic sum of all Delta (respectively all Vega
and Curvatures) calculated on a given risk factor in the trading portfolio. As a
result, sensitivities in opposite directions for a given risk factor compensate
each other, which makes sense from the risk point of view: a position sensitive
to a risk factor in one direction can be hedged by another position that varies
in the opposite direction. In other words, within the same risk class and for
the same indicator, the portfolio's diversification effect is fully exploited to
achieve capital charge "savings" when exposed to market risk.

### Calculate weighted sensitivities

Each net sensitivity is then assigned the risk weight provided in the documentation for the relevant risk factor + risk bucket to give the weighted sensitivity:

### Aggregate sensitivities by risk class

#### Aggregate sensitivities by bucket

Weighted sensitivities by risk bucket must then be aggregated. For Delta and Vega, the formula for calculating the capital charge for bucket b is as follows:

Where:

- `k` and `l` represent risk factors
- `rho_kl` is the correlation parameter between risk factors `k` and `l`
- `WS_k` and `WS_l` are weighted sensitivities to `k` and `l` risk factors

#### Aggregate sensitivities by risk class

The aggregation by risk class is done in a similar way using the results of the previous step and the correlation parameters between buckets within the same risk class:

Where:

- `S_b` (`S_c` respectively) ` = sum_k WS_k` for all risk factors within bucket `b` (respectively `c`)
- `gamma_{bc}` is the correlation parameter between buckets `b` and `c`

#### Remarks

The aggregation formulas for Delta and Vega are the same; they differ slightly for the curvature. The documentation also provides alternative scenarios for the case where the quantity under the square root is negative.

#### Scenarios

This dual aggregation process must be carried out 3 times
per risk class and indicator, with **three different correlation scenarios**:
low, medium and high. For each indicator, we will retain the correlation
scenario with the highest end result (Delta + Vega + Curvature,
for all risk classes).

Using 3 scenarios makes it possible to take into account the fact that in times of market stress, correlations between risk factors may increase or decrease. The "medium" scenario corresponds to the "basic" values of 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.

### Calculate the final capital charge

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

In addition to this capital charge for market risk, which stems directly from sensitivities to risk factors, it will be necessary to add:

- The default risk charge
- The residual risk add-on for residual risk

...in order to obtain the final capital charge: