1. Advanced Risk Management I
Lecture 10
Market risk regulation

2. Market risk regulation: past, present
BIS proposal in1995
Building block approach
Basel 2:
Internal models vs standardized approach
Basel 2.5:
Stressed VaR
IRC and CRM
Basel 3:
CVA

3. RWA
In 1988, the Basel Accord (Basel 1) imposed capital requirements on the assets of banks. The capital had to be 8% of the «risk weighted asset» (RWA). The risk weights were only referred to credit risk.
In 1993 the Basel Committee proposed an amendment to account for market risk. The approach was finalized in 1995.
This substituted the credit risk RWA for the assets in the «trading book».

4. The trading book
The capital requirements for market risk are referred to the «trading book», that is «bank’s proprietary positions in financial instruments which are intentionally held for short term resale and/or which are taken on by the bank with the intention of benefiting in the short term from actual and/or expected differences between their buying and selling prices, or from other price or interest rate changes, and positions in financial instruments arising from matched principal brokering and market making, or positions taken in order to hedge other elements of the trading book»
The counterpart of the «trading book» is the «banking book», that includes positions held for medium/long term investment.

5. The «building block approach»
The capital requirement for market risk is computed as the sum of the capital requirements for the risk factors represented in the «mapped» portfolio:
Interest rate risk
Equity risk
Currency risk
Commidities risk
The capital requirements for interest rate risk and equity risk are the sum of
General risk (market shocks common to all assets)
Specific risk (shocks specific to the issuer)

6. Interest rate risk (specific)
The capital requirements for specific risk substitute those for credit risk imposed in Basel I, for the trading book.
Government bonds with rating from AAA to AA- are given zero weight.
Government bonds with rating between A+ and BBB– and «qualifying issues» absorbe
0.25% (0-6 months)
1.00% (6 – 24 months)
1.60% (more than two years)
8% for any bonds, with rating between BB+ and BB- and unrated bonds
12% for bonds with rating lower than BB-

7. «Qualifying issues» Qualifying issues include:
Bonds issued by public entities other than the state (municipal bonds)
Multilateral agencies (World Bank, EIB,…)
Bonds with investment grade issued by at least to recognized rating agencies, or issued by listed companies that the bank recognizes of comparable quality.

8. Interest rate risk (generic)
Zone
Band
Life to maturity MD r RW 1
Coupon <3%
Duration
Change
MD X r
 1 m.
0.00
1.00%
0.00% 2
1-3 m.
0.20
0.20% 3
3-6 m.
0.40
0.40% 4
6-12 m.
0.70
0.70%
Zone
Band
Life to maturity MD r RW 2 5 y y
1.40
0.90%
1.25% 6 y y
2.20
0.80%
1.75% 7 y y
3.00
0.75%
2.25%

9. Interest rate risk (generic)
Zone
Band
Life to maturity MD r RW
Coupon <3%
Duration
Change
MD X r 8 y y
3.65
0.75%
2.75% 9 y y
4.65
0.70%
3.25% 10 y y
5.80
0.65%
3.75% 11 y
10-15 y
7.50
0.60%
4.50% 3 12 y
15-20 y
8.75
0.60%
5.25% 13 y
Over 20 y
10.0
6.00% 14
12-20 y -
13.50
8.00% 15
21.00
12.50%

10. Netting policies for interest rate risk
Compute the positive and negative net position in each band and compute 10% of the minimum («vertical disallowance»)
Compute the positive and negative net position across band, within each zone and compute c% (40% for zone 1 and 30% for the others) of the the minimum («horizontal disallowance»)
Offset positions across time zones, with allowances 40% if the zones are adjacents and 100% otherwise.

11. Capital requirement for interest rate risk
The capital requirement is computed as the sum of
Vertical disallowances
Horizontal disallowances
Disallowances across zones
The net weighted position across zones.

12. Derivatives
Linear derivatives are mapped on their replicating portfolio.
Non linear derivatives are alternatively represented with
«Delta plus» method: requirements for delta, gamma and vega
«Scenario analysis» method: grid of scenarios of rates and volatilities.

13. Requirements for equity
Specific risk: 8% of the net overall position, that is absolute value of the net overall positions.
Generic risk: 8% of the net overall position, for each market. Derivatives are represented by the delta equivalent.

14. Requirement for forex
The requirement for foreign currencies is obtained computing
The positive net position of all currencies
The negative net position of all currencies
8% of the maximum of the above
It can be proved that this corresponds to the average of perfect dependence and independence.

15. Requirements for commodities
Two alternative ways:
Simplified method: 15% of the net position (long less short position) plus 3% of the gross position (long plus short position).
«Maturity ladder»: the positions are reported among maturity bands, like for interest rate risk. Then, there is a mechanism of disallowances plus the net overall position.

16. Internal models
Since the 90s, large banks had developed sophisticated models for the measurement of market risk.
So, the Committee allowed the possibility to use internal models for the valuation of risk. They required however
Daily estimation of VaR
Confidence level at 99%
Unwinding period 10 days
Historical estimation at least one year
Vol and correlations updatedleast quarterly
Sum of the VaR of different risk factors
Adequate report of non linear positions

17. Organization issues
For the implementation of internal models, banks were required to:
have independent RMU
perform regular stress test for extreme scenarios
backtesting
top management involvement
integration with daily risk management and internal trading limits
approval by supervisory authorities

18. The internal model VaR
In the first implementation of the internal model for market risk the capital requirement was set equal to the sum of:
maximum of previous day VaR and the 60 day average of VaR multiplied times a factor between 3 and 4
the capital requirement for specific market risk

19. Basel 2
Basel 2, in 2004 represented the major change in the regulation, but almost did not touch upon market risk.
Pillar 1: capital requirements for credit risk, market risk, operational risk
Pillar II: supervision from regulatory authorities, and risks not covered in Pillar I
Pillar III: disclosure to the market.

20. Basel 2.5
In 2009 the Basel Committee revised the market risk approach, following the fact that the VaR measure had not worked properly in the crisis
Stressed VaR: it was introduced the requirement of computing the VaR in a stress situation (12 month worst case)
Incremental Risk Charge (IRC): capital allowance for default and rating downgrading in the portfolio
Comprehensive Risk Measure: capital allowance for default, downgrading and price changes of securitization bonds (correlation products)

21. Internal models today
Today, banks using internal models must post capital requirements to cover the sum of:
The higher of i) the previous day VaR number and ii) 60 day average of VaR multiplied by a factor
The higher of i) the latest Stressed VaR number and ii) 60 day average of Stressed VaR multiplied by a factor
IRC (for bonds and bond portfolios)
CRM (for credit correlation products)

22. Other capital requirements
CVA: it was introduced by Basel 3, with a simplified approach and a formula for banks using the internal model.
AVA: requirements for balance sheet opacity and «accounting risk», with a simplified approach equal to 0.1% of the fair value (if lower than 15 billions) or the sum of 9 Additional Valuation Adjustments for Market Price Uncertainty, Closing-out costs, Model risk, Unearned credit spread, Funding value adjustment, Concentration risk, future administrative costs, early termination, operational risk.

23. The future FRTB
New rules for the limitation of trading book and banking book. Criteria will be set by regulatory authorities rather than subjective choices. Changes are made more difficult, subject to approval, and must not decrease the amount of capital.
Standard approach: will be revised including sensitivities to GIRR, spread risk, default risk, share price risk, currency and commodity risk. The sensitivities for options must include «curvature» and vega risk.

24. The future FRTB (continued)
Internal model:
Computed at desk level
Uses expected shortfall calibrated in stress periods
Liquidity horizons will be differentiated by risk factor
Correlations will be defined by the regulation
Incremental default risk (IDR): substitutes IRC. It will be more relevant. It will also apply to stocks.

25. The FED view: stress tests
Apart from risk measures that are used to define capital requirements in Europe, a new doctrine is used in the US.
The FED sets capital requirements using stress tests. Stress tests are also used by ECB, but not for the definition of capital requirements on a current basis.
We can however show that «stress tests» are risk measures themselves.

26. Stress testing
Stress testing techniques allow to evaluate the riskiness of the position to specific events
The choice can be made
Collecting infotmation on particular events or market situations
Using implied expectations in financial instruments, i.e. futures, options, etc…
Scenario construction must be consistent with the correlation structure of data

27. Stress testing How to generate consistent scenarios
Cholesky decomposition
The shock assumed on a given market and/or bucket propagates to others via the Cholesky matrix
Black and Litterman
The scenario selected for a given market and/or bucket is weighted and merged with historical info by a Bayesian technique.

28. Multivariate Normal Variables
Cholesky Decomposition
Denote with X a vector of independent random variables each one of which is ditributed acccording to a standard normal, so that the variance-covariance matrix of X is the n  n identity matrix Assume one wants to use these variables to generate a second set of variables, that will be denoted Y, that will be correlated with variance-covariance matrix given .
The new system of random variables can be found as linear combination of the independent variables
The problem is reduced to determining a matrix A of dimension n n such that

29. Multivariate Normal Variables
Cholescky Decomposition
The solution of the previous problem is not unique meaning that there exost many matrices A that, multiplied by their transposed, give  as a result. If matrix  is positive definite, the most efficient method to solve the problem consists in Cholescky decomposition.
The key point consists in looking for A in the shape of a lower triangular matrix .

30. Multivariate Normal Variables
Cholesky Decomposition
It may be verified that the elements of A can be recoverd by a set of iterative formulas
In the simple two-variable case we have

31. Black and Litterman
The technique proposed in Black and Litterman and largely used in asset management can be used to make the scenarios consistent.
Information sources
Historical (time series of prices)
Implied (cross-section info from derivatives)
Private (produced “in house”)

32. Views
Assume that “in house” someone proposes a “view” on the performance of market 1 and a “view” on that of market 3 with respect to market 2.
Both “views” have error margins i with covariance matrix 
e1' r = q1 + 1
e3' r - e2' r = q2 + 2
The dynamics of percentage price changes r must be “condizioned” on views “view” qi.

33. Conditioning scenarios to “views”
Let us report the “views” in matrixform
and compute the joint distribution ~

34. Conditional distribution
The conditional distribution of r with respect to q is then
and noticed that this may be interpreted as a GLS regression model (generalised least squares)

35. Esempio: costruzione di uno scenario
Assumiamo di costruire uno scenario sulla curva dei tassi a 1, 10 e 30 anni.
I valori di media, deviazione standard e correlazione sono dati da

36. A shock to the term structure

37. Stress testing analysis (1) The short rate increases to 6% (0.1% sd)

38. Stress testing analysis (1) The short rate increases to 6%(1% sd)