RAROC framework, integration and stress testing

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RAROC framework, integration and

stress testing

Ludger Overbeck,

University of Giessen

Risk Management Workshop Colombia: From Theory to Implementation

Agenda

We will consider the following questions:

What is economic capital and RAROC?

Benefits of the RAROC-calculations? Tools and Application.

Measurement of EC

How can the risk types be integrated?

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Cartagena/Columbia· February 2004 · Integration RAROC - · Page 3

Why risk measurement / management?

Success

Successful firms attract more capital

Risks

Most businesses go along with risk taking

Capital

Risks has to be cushioned by capital

Capital and Success can be quantified also risks have to be quanitified!

What is economic capital?

EC is the capital needed as a cushion against large losses.

In mathterms: – Usually:

Quantile of a loss distribution minus its expected loss, Quantile (99%)(L) -EL

– Alternative:

Expected Shortfall: E[L|L>”Large”], possibly “Large”=Quantile.

Can be viewed as a insurance or risk premium, that

conceputually should be invested in riskless and liquid assets! Quantile or “Large” indicated the risk appetite of the institution and depends also on the desired own default probability.

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Loss Distribution

Set Time Horizon Obtain Loss Distribution Set Level of Confidence, e.g. 99%-Quantile Read off Economic Capital 0 0.005 0.01 0.015 0.02 0.025 0.03 0 50 100 150 200 250

Expected Loss Economic Capital

Unexpected Loss

Loss Probability Density or Frequency of

Losses

99%-Quantile

Mean

What is economic capital?

Specification of risk and therefore of loss distribution is necessary. Risk Types: Credit Risk Market Risk Operational Risk Business Risk (?) Liquidity Risk (?) Reputational Risk (?) Legal (?)

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EC and RAROC

Performance measure should take into account risk/return relations

Performance measure should measure the “return” per unit of “risk”

Specification

“Risk”=EC

“Return”=Profits-Costs,

EC and RAROC

RAROC= Risk adjusted Return over “Capital” “Capital”=Risk Capital=Economic Capital Risk Adjusted Return=

Return-Costs (including ”Risk Costs”) “Risk Costs”=Expected Loss

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Expected Loss

For the entire portfolio it equals the expected value= mean value of the loss distribution

Can be calculated “bottom-up” from the single transactions (since mean values and average are additive (as are losses)) The mean value of the loss in a single transaction is then the product of the three mean values of the Loss Given Default-variable the default Default-variable and the Exposure at Default variable

It is usually assumed that the components of EL, namely Loss-Given-Default, Default event and Exposure-At-Default are independent or fixed and non-random.

The abbreviations LGD, EAD usually denote the mean value, whereas the mean value of the loss variable equals the probability of default PD

Expected Loss

EL = PD x LGD x EAD

$$ % % $$

Expected Probability Loss Given Default Exposure at default

Loss of Default

One-year Default Probability_{One-year Default Probability}

Expected loss quota at default (0 < LGD < 1)Expected loss quota at default_{(0 < LGD < 1)}

Expected Exposure at defaultExpected Exposure at _default

Probability of default within one-year

- Definition of default might

depend on counterparty/product

driving factors:

Creditworthiness of counterparty Ratings

Historical loss experience

Loss Given Default

- Percentage of EAD which actually gets lost in case of

default

driving factors:

Collateral Guarantees Product type

Driving factors:

product type market data time to maturity

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Expected Loss

Transaction with

50% loss given default

– Collateral = production engine

Exposure-at-default= 1,500, 000 USD

– Committed line of 2,000,000

– Current usage 1,000,000

– Assumed utilization of undrawn amount of 50%.

Counterparty

PD=0.30%

– Rating=BBB

EL=0.003*0.5*1,500,000 USD= 2,250 USD or in percentage of EAD: EL=0.15%

Economic Capital

The actual calculation of EC is more involved and presented later in the presentation.

In addition to the EL parameters LGD,EAD,PD also dependencies/correlations have to be specified. EC is by its very nature a portfolio characteristic. The breakdown of the portfolio EC to subportfolios and divisions and finally to each single transaction is called Contributory Economic Capital. It is a kind of marginal EC.

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Application of RAROC

RAROC Pricing Tool

Used in lending business for margin calculation

In the trading used for mispricing and information on credit risk capital

Predeal checking

Adds all contributory EC´s.

Only Credit EC is calculated under the full portfolio information.

Other risk types are estimated on individual level

Important other parameter:

Cost function

Application of RAROC

Sensitivity of EC with respect to risk factors like country and industry.

Given a choice of investment yielding the same return and same Expected Loss, the country and industry factor should play a decisive role.

EC-Calculator=Marginal Risk Calculator

Can also be used for evaluation of new businesses.

Next slides provide examples. One sees the marginal capital ( Contributory Economic Capital)

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illustrative

-Contributory Economic Capital as a function of industry

EDF: 30 bp R²: 30 % LGD: 50 % CTY: Germany CEC in % Exposure 0.00% 1.00% 2.00% 3.00% 4.00% 5.00% Chemicals Finance

Companies conductors

Semi-Application of RAROC

EC and RAROC example

As in the EL example: PD=30BP, LGD=50%, EAD=1,500,000, EL=2,250

EC=5% (i.e. Chemical) of exposure=75,000

Assume return (after non-risk cost) of 10,000=0.66% net margin

RAROC=(Return-EL)/75,000=7,750/75,000=10.33%

If we could made the same loan in “semi-conductor” industry,

EC=2.5%, the RAROC would double to 20.33%

A “RAROC-hurdle”-rate of 20 % would only be reached by the second transaction.

The transaction with the “chemical”, could perhaps sold in the market, swapped or syndicated.

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Application of RAROC

0.00% 1.00% 2.00% 3.00% 4.00% 5.00%

Germany USA/Carib. Japan

CEC in % Exposure

Contributory Economic Capital as a function of country

EDF: 30 bp R²: 30 % LGD: 50 % IND: Automotive illustrative

-Application of RAROC

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Measurement of EC

EC in the RAROC formula is contributory or marginal economic capital of the transaction (CEC)

Normally, CEC is calculated separately for each risk type

Most implemented models add the EC obtained from their Market, Credit and Operational Risk calculations Conservative approach “Correlation=1”.

Techniques for measuring the different risk types are broadly the same

Value At Risk Market Risk Market Volatility CreditVaR Credit Risk Defaults of Counterparties Operational Risk Operational Events Operational VaR Aggregation

Measurement of EC

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Concepts for Integration/Aggregation

Loss Distribution for single risk types is specified! How works the aggregation?

Two concepts:

“Additivity” – Top Down Full “Integration”- Bottom up

Aggregation of Risk Types / Top Down

If the loss distributions are separated and the total loss is the sum of the individual loss distributions then we have additivity of risk types.

In formulas

L(total)=LCR+LMR+LOR

If addionally all risk types are driven by factors F(1),..,F(K) the dependence is driven by these factors and integration is straightforward.

In the simulation evaluate LCR, LMR and LOR on each scenario sc

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Aggregation of Risk Types / Top-down

Realistic Example: Generated a two-dimensional loss

distribution (MR,CR) with normal copulawith ρ=0.5

MR=N(0,0.20) (Normal distribution with 20%

Volatility, CR=Vasicek (30bp, 0.12) (see next slides) Can be thought of three standard normal factors

General risk factor f(1)

Idiosyncratic Market risk Factor f(mr) Idiosyncratic Credit Risk Factor f(cr)

F(1)=0.70 f(1)+0.3 f(cr) F(2)=0.70 f(1)+0.3 f(mr)

LCR=LCR(F(1)), LMR=LMR(F(2))

Vasicek Distribution

Infinite granular portfolio by Gordy, also used in Basel II proposals

All obligors same pairwise correlation, R-squared, and same default probability and same EAD*LGD

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0 0.2 0.4 0.6 0.8 1

. Average systematic risk

Low systematic risk

[ 10 % ]

[ 1 % ]

EL

EC(2BP) = 0.51 % of Exp.

EC (2BP)= 4.00 % of Exp.

High systematic risk [ 30 % ] EC (2BP)= 16.38 % of Exp.

systematic risk 1 % 10 % 30 % 99.98%-quantil 0.81 4.30 16.68 EC 0.51 4.00 16.38 UL 0.09 0.35 0.86 Cap. Mult. 5.67 11.43 19.05

Vasicek Distribution

Top-Down Integration: Results

MR: 20%-Vola, 100 Notional , EC(99%)=46

CR: EL=30bp, Correlation 12%, 10000 Notional, EC(99%)=162 Undiversified (i.e. 100% correlation) =208 EC

5% 196 75% 13% 180 25% 10% 187 50% Benfit Diversified EC Correlation

Low benefit since CR dominates anyway. If CR-EC and MR-EC are

of similar size then benefit with 50% correlation equals 16%and

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Bottom-up Integration

Each single transaction carries all risk types

Simplified Example

Consider Derivative Position with value T with counterparty C Change in equity prices change T as well as default probability of C.

Scenario: C is downrated but T increased, overall loss or profit? CR: Increase of T and the increase of PD increased the overall credit risk.

MR: Profit!

Total: Loss or Profit??

Bottom-up Integration

Simplified Example: Possible Answer:

Credit Adjusted Prices “T_cap=(1-PD)*T”

In case of default “PD=1”, I.e. T_cap=0, In case of migration “PD” changed.

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Full Integration

A first step in this “bottom-up” integration is the modeling of “volatile exposure” in credit risk models Credit risk portfolio models as presented usually assume that the Exposure at Default (EAD) is known and deterministic.

Implicitly, it is often assumed that EAD is independent of the default event.

Of course the simple example showed that this is not the case.

Bottom-up approach difficult to obtain and probably also difficult to manage?

Practical Integration and Allocation

For the overall capital it is useful to get an integrated view on all risk types, because of possible

diversification effects.

Diversification benefit reduces all risks and transactions by the same factor (i.e. 10% in the example)

Allocation on single transaction could and is done separately for each risk type.

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Correlation

In credit risk a possible approach for correlations are asset or ability-to-pay correlations

Usually derived from equity ( and balance sheet ) data for listed corporates

Extrapolation to private firms

Statistical analysis for retail customers, e.g. from default rate volatility

Correlation

Two APP paths

APP of firm A APP of firm B correlation • 2 PD • 2 APP-Distributions at horizon correlation •JDP (joint PD ) • joint APP-Distr.

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Correlations

Reduction by factor models

-Firm A Firmb B

Factor Firms A and B are correlated

corr > 0

Common economic factors.

A and B are correlated since they are exposed to the same or correlated factors

EX: BMW and DaimlerChrysler are correlated via Automotive and Germany

corr > 0 corr > 0 • Decomposition of APP-returns - systematic - specific • Decomposition of systematic - Country - Industry

Φ(DaimlerChrysler) = + 0.70 x Φ(Automotive) +0.30xΦ(Airplane)

+ Φ(Germany)

Factor Model

Ψ + Ψ = Φi wic c wij j Firm Risk Firm Specific Risk Systematic Risk Factor ΦΦΦΦ Industry Country Risk Risk 60 Factors ΨΨΨΨi 35 Factors ΨΨΨΨc Industry

Specific Risk Specific RiskCountry

Global Economic Risk Regional Risk 14 Factors Industrial Sector Risk η i ε

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Threshold Models

Example: Yields a correlation of 30%:

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Financials GER

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2 2 1 GER USA Automotive Financials 1

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Loss Distribution of a Credit Portfolio

Example of Loss Distribution from MonteCarloSimulation

-after last simulation

Generation of asset return for all counterparties

correlated via the factor model No default Default Add exposure of counterparty to loss Next counterparty Portfolio loss in simulation Next simulation after last counterparty = = < N k m i APP C i k x l i i 1 [0, ] 1 { } 1 1 on distributi loss empirical

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Economic Capital

Expected Loss is now the average over all scenarios of the simulated losses

Economic Capital at level 99% for example is now the 1000th largest loss if 100 000 scenarios were generated.

In this scenario generating approach the factors can be re-directed according to a pre-described scenario (see stress testing)

RAROC/EC

Summary

RAROC is an adequate tool Consistent Risk/Return relation

Diversification benefits versus Concentration risk Integration is possible

Top-down, correlation between risk types Bottom-up, all risk types in a single transaction

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Stress-Testing

Scenarios can be defined in terms of the underlying factor model

Downturn in Germany factor (= German economy) by 10% Downturn of automotive by 15%

These scenarios imply a new distribution of the whole factor model, namely the distribution of the factors under the conditions which are formulated in the scenarios

Since the multivariate distribution of the factors is specified, also the conditional distributions are given

Portfolio Level Stress-Testing

It is of course possible and reasonable to define a stress scenario with a set of conditions

The stress scenarios define though in a first step a new distributions of the factor model

In the second step the distribution of L_pthe possible

losses under each of this stress scenarios is derived Therefore each stress scenarios requires a new economic capital for credit risk

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Germany -28% (1 out of 10 years case)

Scenario 1: downturn in German factor

U.S. -24% Business prod. whlsl. -20%

Germany -28% Consumer durables -14%

Consumer products -23% Banks -11%

Scenario 2: global recession, indicated by

Examples

DAXandAutomotivemove quite in parallel under normal conditions

Stress on DAX, below -28% Stress on DAXinfluences

Automotivenegatively. Its

distribution is also pushed down

Impact on extreme losses*

Stress Testing - Illustration

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Stress Testing Results: EC

-illustrative-Stress Testing Results: EL

illustrative

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Stress Testing Results

Summary:

Stress influences Expected Loss more strongly than EC In the EC, there are already the stress scenarios considered, however with a different “the normal” probability weighting Global crisis are much worse in EL terms

Consistent measurement of stress scenarios possible Change in correlation structure only implicit, not explicit. Useful management information:

What happens with capital basis if crisis occurs? Testing of stability of financial system?