RAROC framework, integration and stress testing

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? Stress testing within the EC-framework Cartagena/Columbia February 2004 Integration RAROC - Page 2

Why risk measurement / management? Successful firms attract more capital Success Most businesses go along with risk taking Capital Risks has to be cushioned by capital Risks Capital and Success can be quantified also risks have to be quanitified! Cartagena/Columbia February 2004 Integration RAROC - Page 3 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. Cartagena/Columbia February 2004 Integration RAROC - Page 4

Loss Distribution Probability Density or Frequency of Losses Set Time Horizon Obtain Loss Distribution Set Level of Confidence, e.g. 99%-Quantile 250 200 150 Unexpected Loss Read off Economic Capital 100 50 Mean 99%-Quantile 0 0 0.005 0.01 0.015 0.02 0.025 0.03 Loss Expected Loss Economic Capital Cartagena/Columbia February 2004 Integration RAROC - Page 5 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 (?) Cartagena/Columbia February 2004 Integration RAROC - Page 6

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, Cartagena/Columbia February 2004 Integration RAROC - Page 7 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 RAROC=RAR/EC Cartagena/Columbia February 2004 Integration RAROC - Page 8

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 Defaultvariable the 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 Cartagena/Columbia February 2004 Integration RAROC - Page 9 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 Probability of default within oneyear - Definition of default might depend on counterparty/product driving factors: Creditworthiness of counterparty Ratings Historical loss experience Expectedloss quotaat at default (0 (0 < LGD LGD < 1) 1) Loss Given Default - Percentage of EAD which actually gets lost in case of default driving factors: Collateral Guarantees Product type Expected Exposureat at default Driving factors: product type market data time to maturity Cartagena/Columbia February 2004 Integration RAROC - Page 10

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% Cartagena/Columbia February 2004 Integration RAROC - Page 11 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. Cartagena/Columbia February 2004 Integration RAROC - Page 12

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 ś. Only Credit EC is calculated under the full portfolio information. Other risk types are estimated on individual level Important other parameter: Cost function Cartagena/Columbia February 2004 Integration RAROC - Page 13 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) Cartagena/Columbia February 2004 Integration RAROC - Page 14

Application of RAROC - illustrative - Contributory Economic Capital as a function of industry EDF: 30 bp R²: 30 % LGD: 50 % CTY: Germany CEC in % Exposure 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% Chemicals Finance Companies Semiconductors Cartagena/Columbia February 2004 Integration RAROC - Page 15 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. Bank has to much concentration in chemical Cartagena/Columbia February 2004 Integration RAROC - Page 16

Application of RAROC - illustrative - Contributory Economic Capital as a function of country EDF: 30 bp R²: 30 % LGD: 50 % IND: Automotive CEC in % Exposure 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% Germany USA/Carib. Japan Cartagena/Columbia February 2004 Integration RAROC - Page 17 Application of RAROC Cartagena/Columbia February 2004 Integration RAROC - Page 18

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 Cartagena/Columbia February 2004 Integration RAROC - Page 19 Measurement of EC Market Risk Credit Risk Operational Risk Market Volatility Defaults of Counterparties Operational Events Value At Risk CreditVaR Aggregation Operational VaR Economic Capital Cartagena/Columbia February 2004 Integration RAROC - Page 20

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 Cartagena/Columbia February 2004 Integration RAROC - Page 21 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 L(total,sc)=LCR(f(1,sc),..f(K,sc))+LMR(f(1,sc),..f(K,sc))+LOR(f(1,sc),..f(K,sc)) Cartagena/Columbia February 2004 Integration RAROC - Page 22

Aggregation of Risk Types / Top-down Realistic Example: Generated a two-dimensional loss distribution (MR,CR) with normal copula with 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)) Cartagena/Columbia February 2004 Integration RAROC - Page 23 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 Infinite granular means no idiosyncratic risk left Cartagena/Columbia February 2004 Integration RAROC - Page 24

Vasicek Distribution Low systematic risk [ 1 % ] EC(2BP) = 0.51 % of Exp. Average systematic risk [ 10 % ] 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 0 0.2 EL 0.4 0.6 0.8 1. Cartagena/Columbia February 2004 Integration RAROC - Page 25 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 Correlation 50% 25% 75% Diversified EC 187 180 196 Benfit 10% 13% 5% Low benefit since CR dominates anyway. If CR-EC and MR-EC are of similar size then benefit with 50% correlation equals 16% and with 25% correlation diversification benefit is 27% Cartagena/Columbia February 2004 Integration RAROC - Page 26

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?? Cartagena/Columbia February 2004 Integration RAROC - Page 27 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. Cartagena/Columbia February 2004 Integration RAROC - Page 28

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? Cartagena/Columbia February 2004 Integration RAROC - Page 29 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. Cartagena/Columbia February 2004 Integration RAROC - Page 30

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 Cartagena/Columbia February 2004 Integration RAROC - Page 31 Correlation APP of firm A correlation Two APP paths APP of firm B 2 PD 2 APP-Distributions at horizon correlation JDP (joint PD ) joint APP-Distr. (bivariat normal) today horizon Cartagena/Columbia February 2004 Integration RAROC - Page 32

Correlations - Reduction by factor models - Firms A and B are correlated corr > 0 Firm A Firmb B Decomposition of APP-returns Φ(DaimlerChrysler) = + 0.70 x Φ(Automotive) +0.30xΦ(Airplane) - systematic corr > 0 + Φ(Germany) corr > 0 - specific Factor Common economic factors. Decomposition of systematic - Country - Industry 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 Cartagena/Columbia February 2004 Integration RAROC - Page 33 Factor Model Firm Risk Φ = i w Ψ + ic c w Ψ ij j Systematic Risk Factor Φ Firm Specific Risk ε i Industry Risk Country Risk 60 Factors Ψ i 35 Factors Ψ c Industry Specific Risk Country Specific Risk η Global Economic Risk Regional Risk 14 Factors Industrial Sector Risk Cartagena/Columbia February 2004 Integration RAROC - Page 34

Threshold Models Example: A = 0.6 1 A = 0.5 2 ( 0.3 GFinancials + 0.7 GAutomotive + 0.5 GUSA + 0.5 GGER ) ( GFinancials + GGER ) + ε2 + ε 1 Yields a correlation of 30%: Cartagena/Columbia February 2004 Integration RAROC - Page 35 Loss Distribution of a Credit Portfolio - Example of Loss Distribution from Monte-Carlo-Simulation - Generation of asset return for all counterparties correlated via the factor model No default Next counterparty Portfolio loss in simulation Default Add exposure of counterparty to loss after last counterparty empirical loss distribution N m k 1 [ l i1 0, x] k = 1 i= 1 { APPi < C } i Next simulation after last simulation Loss distribution Example: Portfolio of 2.000 middle market loans Cartagena/Columbia February 2004 Integration RAROC - Page 36

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) Cartagena/Columbia February 2004 Integration RAROC - Page 37 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 Cartagena/Columbia February 2004 Integration RAROC - Page 38

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 Cartagena/Columbia February 2004 Integration RAROC - Page 39 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 p the possible losses under each of this stress scenarios is derived Therefore each stress scenarios requires a new economic capital for credit risk Cartagena/Columbia February 2004 Integration RAROC - Page 40

Examples Scenario 1: downturn in German factor Germany -28% (1 out of 10 years case) Scenario 2: global recession, indicated by U.S. -24% Business prod. whlsl. -20% Germany -28% Consumer durables -14% Consumer products -23% Banks -11% Cartagena/Columbia February 2004 Integration RAROC - Page 41 Stress Testing - Illustration DAX and Automotive move quite in parallel under normal conditions Stress on DAX influences Automotive negatively. Its distribution is also pushed down Impact on extreme losses* Stress on DAX, below -28% Cartagena/Columbia February 2004 Integration RAROC - Page 42

Stress Testing Results: EC -illustrative- Cartagena/Columbia February 2004 Integration RAROC - Page 43 Stress Testing Results: EL illustrative Cartagena/Columbia February 2004 Integration RAROC - Page 44

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? Cartagena/Columbia February 2004 Integration RAROC - Page 45