Presentation, Interpretation, and Use of Stress Test Results

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1 Presentation, Interpretation, and Use of Thomas Breuer Advanced Stress Testing Techniques RISK Training London, 18 May, 2007

2 Agenda Action triggered

3 Value at Risk is not coherent Bank VaR-limit Desk A 70m EUR Desk B VaR-limit 50m EUR 20m EUR Portfolio short 1m Eur. Puts on equity strike: 9200 short 1m Eur. Calls on equity strike: equity currently , r=5%, vola=5%, ttm=5mths Portfolio VaR 42.92m EUR 18.47m EUR Bank VaR 80.91m EUR Breuer 2003

4 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

5 Traditional Stress Testing Stress testing does not share these shortcomings of VaR. Traditional ways to find scenarios But...

6 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

7 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

8 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

9 Choice of scenarios is critical in performing stress tests Understanding scenarios is critical in understanding stress test results 1. How can we communicate a scenario? 2. How should we report? 3. How can we derive action?

10 Risk Factors Market risk factors: stock and commodity prices, exchange rates, interest rates Credit risk factors: RiskMetrics: rating of each obligor, common factors KMV: EDF of each obligor, common factors CreditRisk+: number of defaults in exposure bands, background factors CreditPortfolioView: macroeconomic variables

11 Scenarios: Full vs Partial Full Full stress scenario: specifies values of all risk factors Partial stress scenario: specifies values of some risk factors, usually one or two. From partial scenario to full scenario: Fill in values of missing risk factors, e.g. current value (less plausible), conditional expectation given values of specified risk factors (more plausible)

12 Scenarios: Others Deterministic scenario: A state s of the world at the time horizon, as far it is of relevance to the portfolio value. Probabilistic scenarios: probability dist over deterministic scenarios. Not used here. Multi-step scenarios: not considered here.

13 Traditional Stress Tests with Full Scenarios 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,...}

14 Traditional Stress Tests with Partial Scenarios Partial scenario specifies not a unique portfolio value, but a distribution of values. Choose criterion C to compare distributions: e.g. expectation value, risk measure (variance, quantile, expected shortfall,...) Performing stress tests: 1. Select partial scenarios s stress1, s stress2, Calculate conditional distributions of portfolio values P s stress1, P s stress2, 3. Calculate comparison criterion for cond. distributions C(P s stress1 ), C(P s stress2 ), 4. Compare these values with criterion for uncond. distribution C(P)

15 New Stress Testing: Find Worst Case Scenarios Worst Full Scenario: The full scenario in predefined set of scenarios which has maximum loss Worst Partial Scenario: The partial scenario (with selected risk factors in some predefined ranges) with worst value of criterion C Plausibility: should be taken into account when defining set of scenarios

16 MaxLoss Contribution of risk factor i: (Loss if RF i has its worst case value and other RF unchanged) / MaxLoss Report either worst partial scenario, or key risk factors of full scenarios, but not full worst case scenario only by a few risk factor values, namely those with highest Maximum Loss contribution

17 Report Key Risk Factors of Worst Case MaxLoss contribution of risk factor i: (Loss if RF i has its worst case value and other RF unchanged) / MaxLoss

18 Visualise Portfolio Behaviour in Dependence of Key Risk Factors current USD.SE: 1091 current EUR.Z10: 0.053

19 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 specified by a risk measure

20 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

21 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

22 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.

23 Capital Allocation 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?

24 Capital Allocation 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.

25 Requirements for Capital Allocation 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

26 Risk Measures vs. Capital Allocation 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 i, P ) := lim a 0 r(p + ap i ) r(p ) a Cptl allocation RC ( is linear, diversifying if and only if risk meas. r is sub-additive, homogeneous Kalkbrener 2005

27 Synthesis: risk management Given a set AD of scenarios we get 1. Risk measure r: ( loss in worst scenario r(p ) = MaxLoss AD (P ) = max [P (s CM) P (s)] s AD ) 2. Capital allocation to subportfolios P 1, P 2,, P n of P RC(P i, P ) P i (s CM ) + 1 a for some a >0. ( ) min [P (s)] min [P (s) ap i(s)] s AD s AD Capital allocated to P i = Current Value of P i plus 1/a times Maximum Loss change caused by deducing ap i from P.

28 Example: Risk Allocation with a few prespecified stress scenarios a=0.001 Portfolio P_CM Stress Loss 1 Stress Loss 2 Stress Loss 3 Risk Cptl For very small a, for all subportfolios we have: loss (totalportf-wcs) = RC (subportfolio) loss (subportfolio-wcs) RC (subportfolio) will be almost equal to Loss (totalportf-wcs) RC of total portfolio = sum of RC over subportfolios Lower total RC, no diversification effects for RC Risk Cptl Total Portf Bonds Europe 6, ,700 1,500 1,500 Consumer 3, ,000 1,300 1,300 Energy & Utilities 4, ,600 1,600 Financials 5, ,300 1,300 Healthcare 2, Industrials 5, ,300 1,300 Technology 1, ,400 1,400 Total 29, ,050 9,100 9,100 9,100 RC(subportfolio) only sensitive to totalportf-wcs, not to subportfolio-wcs

29 Example: Risk Allocation with a few prespecified stress scenarios a=0.1 Portfolio P_CM Stress Loss 1 Stress Loss 2 Stress Loss 3 Risk Cptl For not so small a, for all subportfolios loss (portfolio-wcs) RC (subportfolio) loss (subportfolio-wcs) RC (subportfolio) will be close to Loss (subportfolio-wcs) Risk Cptl Total Portf Bonds Europe 6, ,700 1,500 7,200 Consumer 3, ,000 1,300 1,300 Energy & Utilities 4, ,600 1,600 Financials 5, ,300 1,300 Healthcare 2, Industrials 5, ,300 1,300 Technology 1, ,400 1,400 Total 29, ,050 9,100 14,800 9,100 RC of total portfolio sum of RC over subportfolios Larger RC of subportfolios, diversification effects for RC

30 : Reduce exposure to key risk factors Remember: Identify key risk factors

31 : Reduce exposure to key risk factors Remember: Portfolio behaviour in dependence of key risk factors current USD.SE: 1091 current EUR.Z10: 0.053

32 Insurance Position Take up insurance with pay-off precisely in worst case: Pays USD if USD.SE around Fair price of insurance: 349 USD

33 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 puts puts calls calls 1000

34 Visualise Insured Portfolio Behaviour original portfolio portfolio insured

35 Report on Insured Portfolio Insured Portfolio Portf. value Current Mkt. State ,53 Worst Case Sc ,01 MaxLoss ,52 Original Portfolio Portf. value Current Mkt. State ,41 Worst Case Sc ,62 MaxLoss ,786 An insurance of 349 USD reduces MaxLoss by USD.

36 And here are the four main points again 1. Measuring Risk Unlike VaR risk measure r(p ) = MaxLoss AD (P ) = max s AD [P (s CM) P (s)] is a safe basis of firm-wide risk managament.

37 Do not choose a too small. And here are the four main points again 2. Capital Allocation 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. )

38 The four main points again 3. : Report key risk factors of worst full scenario the risk factors with high loss (if RF i has its worst case value and other RF unchanged) or worst partial scenario. Locating the vulnerable spots of a portfolio

39 The four 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

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