Using Stress Test Thomas Breuer thomas.breuer@fhv.at Advanced Stress Testing Techniques Risk Training London, 25 April, 2006
Agenda Action triggered
Risk Risk of a portfolio is determined by its profit/loss distribution Intuition: Risk of a portfolio = Economic capital needed to hold portfolio This should be measured by a risk measure
Requirements on Risk Measures Coherent risk measures r should satisfy the following for all portfolios P1, P2: : 1. Diversification (sub-additivity): r(p1)+r(p2) r(p1+p2) 2. Scaling: r(a P1)= a r(p1) if a positive 3. Lower portfolio value, higher risk: If P1 P2 almost surely, then r(p1) r(p2) 4. Additional capital reduces risk r(p1 + a) = r(p1) a Artzner Delbaen, Eber, Heath 1998
Scenarios (one step) Deterministic scenario: A state s of the world at the time horizon, as far it is of relevance to the portfolio value. Probabilistic scenarios not necessary. Multi-step scenarios not considered here.
All risk measurement is scenario-based Theorem Assume portfolio value is a continuous function of risk factors. A risk measure r is coherent if and only if there is a family AD of scenarios s such that r(p) = max s in AD (a - P(s)) Artzner et al 1998 Kretschmer 2004 Breuer 2007 Risk of portfolio P equals worst loss in set AD No other risk measures are coherent
Value at Risk is not coherent Desk A Desk B Bank VaR-limit 70m EUR VaR-limit 50m EUR 20m EUR Portfolio short 1m Eur. Puts on equity strike: 9200 short 1m Eur. Calls on equity strike: 11300 equity currently 10.000, r=5%, vola=5%, ttm=5mths Portfolio VaR 42.92m EUR 18.47m EUR Bank VaR 80.91m EUR Breuer 2003
Dangers of Value at Risk VaR of a portfolio might be larger than sum of the VaRs of its sub-portfolios: VaR not safe for firm wide limit system, capital allocation etc. Many implementations of VaR underestimate the probability of extreme events No information about size of losses exceeding VaR No information about dangerous situations
Traditional Stress Tests Current state of the market: s CM Hence, current portfolio value: P (s CM ) Performing stress tests: 1. Select scenarios s stress1, s stress2,... (according to some criterion) 2. Calculate portfolio values P (s stress1 ), P (s stress2 ), 3. Compare these values with P (s CM ) Stress test = scenario analysis AD={s stress1, s stress2,...}
Traditional Stress Tests How to select scenarios Standard scenarios Historical scenarios Subjective worst case scenarios
Dangers of Traditional Stress Tests For a sample portfolio of equities: Stress tests with standard and historical scenarios may nourish a false illusion of safety
Dangers of Traditional Stress Tests A stress scenario for one portfolio might be a lucky strike for another portfolio Stress tests with standard and historical scenarios may nourish a false illusion of safety Subjective worst case scenarios are often too implausible to trigger management action
Dangers of Traditional Stress Tests But: Stress Tests can be the basis of informed risk decisions...... if the scenarios are plausible... if we are confident that there are no worse and more plausible scenarios
Maximum Loss Admissibility domain AD: Scenarios above a certain minimal plausibility threshold For normal or t-dist AD is an ellipsoid. MaxLoss AD (P ) = max s AD [P (s CM) P (s)] Above the plausibility threshold nothing worse than MaxLoss can happen.
Capital Risk-return based incentives for managers of subportfolios / business units How should we split the risk capital to subportfolios / business units? Who gets the benefits of diversification?
Capital RC(P1, P): risk capital of portfolio P1 as a subportfolio of portfolio P RC(P1, P) different from RC(P1, P2) : Different diversification of P1 with the rest of portfolio P than with the rest of P2.
Requirements for Capital Rules 1. RC(aP 1 +bp 2, P) = a RC(P 1, P)+b RC(P 2, P): so RC(total porfolio) = sum of RC of subportfolios 2. RC(P1, P1) RC(P1, P) only positive diversification effects RC of P1 as standalone portfolio is bigger than RC of X as subportfolio of p 3. Continuity in P
Risk Measures vs. Capital Rules One-to-one correspondence of risk measures and cptl allocation rules: risk measure determined from cap allocation rule r(p) := RC(P,P) cap allocation determined from risk meas.: RC(P 1, P ) := lim a 0 r(p + ap 1 ) r(p ) a Cptl allocation ( RC is linear, diversifying if and only if risk meas. r is sub-additive, homogeneous Kalkbrener 2005
Synthesis: Scenario-based risk management Given a set AD of scenarios we get 1. Risk measure r: loss in worst scenario 2. Capital allocation to subportfolios P 1, P 2,, P n of P for some small a ( RC(P i, P ) P i (s CM ) + 1 a r(p ) = MaxLoss AD (P ) = max [P (s CM) P (s)] s AD ( min s AD ) ) [P (s)] min [P (s) ap i(s)] s AD Capital allocated to P i = Current Value of P i plus 1/a times Maximum Loss change caused by adding ap i to P.
Choice of scenarios is critical in performing stress tests Understanding scenarios is critical in understanding stress test results How can we communicate a scenario? (the value of each of several hundred risk factors)
MaxLoss contribution of risk factor i: (Loss if RF i has its worst case value and other RF unchanged) / MaxLoss For reports, characterise scenarios only by values of risk factors with highest Maximum Loss contribution
Identification of Worst Case
Identifying Decisive Risk Factors MaxLoss contribution of risk factor i: (Loss if RF i has its worst case value and other RF unchanged) / MaxLoss
Visualise Portfolio Behaviour current USD.SE: 1091 current EUR.Z10: 0.053
Insurance Position Take up insurance with pay-off precisely in worst case: Pays USD 43.000 if USD.SE around 1025. Price of insurance: 349 USD
Construction of insurance positions If risk factor is traded. Current value of risk factor: 1091 Worst case value: 1018 Daily vola of risk factor: 1,43% # of options Strike 1250 puts 1070-2500 puts 1090 1250 puts 1000-1000 calls 1040 1000 calls 1000
Visualise Insured Portfolio Behaviour original portfolio insured portfolio
Insured Portfolio Insured Portfolio Portf. value Current Mkt. State 403.304,53 Worst Case Sc. 373.845,01 MaxLoss 29.459,52 Original Portfolio Portf. value Current Mkt. State 402.955,41 Worst Case Sc. 330.764,62 MaxLoss 72.190,786 An insurance of 349 USD reduces MaxLoss by 42.731 USD.
And here are the four main points again Introduction Reporting Stress Test Scenario-based Risk and Capital 1. Measuring Risk Unlike VaR Scenario-based risk measure r(p ) = MaxLoss AD (P ) = max s AD [P (s CM) P (s)] is a safe basis of firm-wide risk managament.
Do not choose a too small. And here are the four main points again Introduction Reporting Stress Test Scenario-based Risk and Capital 2. Capital Scenario-based capital allocation to subportfolios P i : for some small a. ( RC(P i, P ) P i (s CM ) + 1 a ( ) min [P (s)] min [P (s) ap i(s)] s AD s AD This capital allocation is consistent and diversifying. )
The 4 main points again 3. Reporting: For reports, identify the key risk factors of the worst case scenario: the risk factors with high loss (if RF i has its worst case value and other RF unchanged) Locating the vulnerable spots of a portfolio
The 4 main points again 4. Action triggered Insuring portfolio against extreme loss: Take counter-positions which are profitable when key risk factors have their worst case values