Columbia FE Practitioners Seminar Hedge Fund Risk

Transcription

1 Columbia FE Practitioners Seminar Hedge Fund Risk March 22, 2010 Yann Coatanlem Managing Director Multi-Asset Quantitative Analysis Citigroup

2 Introduction

3 How did Value-At-Risk perform during the current crisis? Strong perception that most risk models and all VaR models of banks failed It is true that many large banks reporting a few hundred million dollars of VaR to the Federal Reserve and other regulators, ended up with billions of dollars of losses Banks PL exceeded their VaR levels far in excess of the confidence level But Page 3

4 How did Value-At-Risk perform during the current crisis? But this disastrous failure was due to one mistake of historic proportions: treating subprime CDOs as risk-free bonds In most trading departments risk monitoring and control remained perfectly adequate We will take a look at how hundreds of real life hedge fund portfolios and hundreds of dummy portfolios behaved We will show that risk model designed BEFORE the current crisis sustained the various systemic chocks of this crisis fairly well Page 4

5 A Multi-Asset / Multi-Layer Model

6 Cross Product Margining for Hedge Funds Hedge Funds might have ratings (SP, Moody s) But Hedge Funds do not issue bonds, so their credit risk can not be hedged! (A few exceptions: i.e. Citadel) Solution: expect the worst possible outcome at some confidence level Hedge Funds have to post a daily margin including: Mark-to-market changes (Variation margin) n-day, high confidence level (99% or higher) portfolio Value-At- Risk (VaR) or Expected Shortfall (ES) Liquidity spreads => mostly trading desk inputs (i.e. CDOs bid-ask, hedged vs. non-hedged) Page 6

7 Modeling Challenges Large number of market factors: several thousands across Fixed Income, FX, Credit, EM, and MBS For options risk is no longer linear => we need to use Monte-Carlo simulation (and try to avoid full repricing ) A lot of missing data (Credit, EM) => completion algorithm (EMRidge) Time window: Flat VaR (10 years) versus Dynamic VaR (3-6 months) Correlation stressing and fat tail adjustment Multiple VaR components: Market, Default, Liquidity, Concentration, etc Page 7

8 Using different statistical regimes Two types of covariance matrices are generated: Flat matrix: using a back window of equally weighted data accreting from Jan 1997 to today Dynamic matrix: using exponentially weighted data with a decay coefficient of, say, 0.99 The Market VaR is the maximum VaR obtained under these 2 methods. This is to ensure that VaR increases quickly in case of volatile markets (dynamic effect), but decreases slowly when markets are quiet (flat effect) Page 8

9 Adjusting for fat tails We assume that the conditional distribution of market factor changes is normal However, the volatility at time t is adjusted such that the cumulative normal distribution exactly matches the historical distribution at a certain percentile. In effect, we scale the volatility so that the resultant VaR equals the historical VaR (HVaR) To be more conservative, we scale the volatility to the maximum HVaR calculated for both positive and negative moves (i.e. to both sides of the distribution). The fat tail perturbation is done sequentially on overlapping blocks of data. For instance block 1 could include USD Libor, Treasury and Agency, block 2 most foreign Libor and sovereign curves The algorithm starts with block 1, computes its principal components, constructs time series of the principal components, bumps the standard deviation of each principal component to match its historical percentile, and reconstructs the original time series Page 9 Then we move to block 2 and do the same, etc until the last block is done. Some of the blocks are overlapping in order to stress crosscorrelations

10 Market VaR for Credit Products Scope: single name corporate bonds, CDS, Loans, LCDS, CDS/Loans indices, sovereign bonds, CDOs Choose a model which can be calibrated to a cross-section of liquid instruments, i.e. via maximum likelihood estimation Prefer lognormal or shifted lognormal dynamics => more reactive to market moves Need for industry / sector specific factors Need to model all the basis spreads between bonds, CDS, Loans, LCDS Need for idiosyncratic factors Page 10

11 Default C-VaR / ES for Credit Products Need to account for jump to junk more than jump to default: default usually occurs when prices have more or less converged to the recovery value The default VaR for a corporate credit portfolio is taken to be the conditional expected loss due to defaults at a high percentile (say 99.9%): CVaR CVaR is a function of default probabilities for each issuer, the positive net loss given default aggregated by issuer and the sufficient parameters of the copula linking defaults among the issuers; the correlation parameters used are reviewed periodically to ensure they are in line with the base correlation for synthetic equity tranches for CDX and Itraxx investment grade indices The initial calculation of a conditional VaR at the 99.9th percentile confidence interval is motivated by the high skewness of the portfolio loss distribution, given that defaults are by their nature infrequent events on a few days/weeks horizon Page 11

12 Default C-VaR / ES for Credit Products (Con d) Two separate calculations are performed to determine portfolio loss distribution for default risk adjustment The first calculation uses historical long-term default frequencies and recovery rates based on issuers' Senior Unsecured obligations ratings The second uses implied hazard rates obtained from issuers' current short term market CDS levels In both calculations, these are used to establish issuers' individual default probabilities The copula model is then used to compute the conditional expected loss. Numerically the calculation uses the iterative method described in All your hedges in one basket by Andersen, Basu and Sidenius, Risk, November, 2003, pages Page 12

13 Backtesting

14 Constructing Test Portfolios Scope of products: interest rates derivatives, FX forwards and options, sovereign credit, corporate credit About 1,300 hedge fund portfolios and dummy portfolios (long, steepening / flattening, butterflies, vol trades, FX trades, etc) Specific credit strategies: stressed credit names vs. CDXIG8 CDXIG 3y vs. 5y CDXHY 3y vs. 5y Single names Page 14

15 Backtesting results for Market VaR The 1,300 portfolios were backtested over 12 years Model output is compared to 5-day PL changes Over 12 years: The average number of exceptions is 0.5% The average shortfall is 30.6% of model output Over the last 2 years: The average number of exceptions is 0.9% The average shortfall is 32.2% of model output Page 15

16 Backtesting results for Market VaR Hedge Fund Client X $150,000,000 $100,000,000 $50,000,000 $0 Source: Citigroup 7/24/98 12/6/99 4/19/01 9/1/02 1/14/04 5/28/05 10/10/0 6 -$50,000,000 2/22/08 7/6/09 11/18/1 0 PnL Flat- Flat+ Dynamic- Dynamic+ Max- Max+ -$100,000,000 -$150,000,000 Source: Citigroup Page 16

17 Backtesting of 28 stressed credit names vs. CDXIG8 Ambac Assurance Corporation Abitibi Bowater Bear Stearns Beazer Homes Continental Airlines Countrywide Home Loans Clear Channel Countrywide Financial Corporation CIT Group, Inc Ford Motor Credit GMAC General Motors Corporation Hilton Hotels Corporation Health Management Associates Lehman Brothers MBIA Insurance Corporation Metlife Monolines Pulte Homes Six Flags Radian Group, Inc. Residential Capital, LLC SLM Corporation Toll Brothers Tribune Company Washington Mutual, Inc XL Capital Ltd Page 17

18 Backtesting of 28 stressed credit names vs. CDXIG8 28 stressed credits versus CDXIG8 $1,000,000 $500,000 $- 3/30/2007 9/30/2007 3/30/2008 9/30/2008 3/30/2009 9/30/2009 $(500,000) $(1,000,000) $(1,500,000) $(2,000,000) $(2,500,000) MaxMarketVaR+ MaxMarketVaR- TotalVaR- TotalVaR+ PL Page 18 Source: Citigroup

19 Backtesting of Lehman Brothers CDS Lehman Brothers $400,000 $200,000 $- 3/30/2007 9/30/2007 3/30/2008 9/30/2008 3/30/2009 9/30/2009 $(200,000) $(400,000) $(600,000) $(800,000) $(1,000,000) $(1,200,000) MaxMarketVaR+ MaxMarketVaR- TotalVaR- TotalVaR+ PL Page 19 Source: Citigroup

20 Backtesting of WaMu CDS Washington Mutual, Inc $1,000,000 $500,000 $- 3/30/2007 $(500,000) 9/30/2007 3/30/2008 9/30/2008 3/30/2009 9/30/2009 $(1,000,000) $(1,500,000) $(2,000,000) $(2,500,000) $(3,000,000) $(3,500,000) MaxMarketVaR+ MaxMarketVaR- TotalVaR- TotalVaR+ PL Page 20 Source: Citigroup

21 Stress Tests

22 Page 22 Stress Tests Common opinion goes something like this: VaR models can shoot in many directions but often miss their target; stress tests are good at reaching their targets, but a limited number of them and are often more transparent There is, in fact, no contradiction, no fundamental difference between these two approaches: they just focus on different parts of the portfolio tail distribution Conditional stress tests in particular require the same amount of information as VaR models But the key difficulty with stress tests is: what weights apply to them? Quants/rating agencies/regulators knew very well that the US housing market could collapse, but they had very limited means to extract probabilities from market prices

23 Extreme Value Theory

24 EVT Methodology Our univariate model for VaR is portfolio-specific. We use a 2- step method: PL t = σ t Z t 1 st step: estimate conditional volatility σ t : (1,1) σ t2 = a + b σ t-12 + c (PL t-1 ) 2 2 nd step: estimate quantile on Z t ~ i.i.d.(0,1), using various techniques : HS (historical simulation) L-estimators N (Normal): Φ -1 (0.99) EVT (Extreme Value Theory): developed to measure the probability of extreme events various econometric techniques to estimate the Generalized Pareto Distribution: Hill estimator, Maximum Likelihood MLE, Proba-Weighted Moments PWM Page 24

25 Testing the VaR model 1. Backtesting VaR on real PL: compare EVT to current model used for hedge fund margining; and to HS- and L- estimators 2. Backtesting VaR on returns on SP, Foreign Currencies, Commodities Page 25

26 Backtesting of Real Portfolios Assessment of each VaR model, based on: Exceptions Losses upon exception A measure of ES fit Implemented for 198 actual portfolios, over the last 10 years: Available history starts on Jan. 9, 1997 We estimate VaR and Expected Shortfall (ES) from May 1, 2001 until July 11, 2007 We compare VaR 99 against PL for 1,630 dates: we therefore expect ~16 exceptions Page 26

27 A Measure of ES Fit, For Each Portfolio Ε [ Loss VaR Loss > VaR] ES VaR = 1 ES fit i = T max ( 0, Loss VaR ) t= 1 i, t i, t T 1[ Loss > ] i, t VaRi, t t= 1 ES i, t VaR i, t Page 27

28 Summary: ES fit, Exceptions, Losses HS L N EVT-Hill EVT-MLE EVT-PWM ES fit (should be =1) Mean # Exceptions over PnL (should be 16) Mean (over PnL) [% Loss over VaR / T] Mean (over PnL) [% Loss over VaR / #exceptions] Source: Citigroup Page 28

29 Backtesting results for Market VaR VaR for all portfolios, on July 11, 2007 $1,000,000,000 $100,000,000 $10,000,000 EVT $1,000,000 $100,000 $10,000 $1,000 $100 $10 $1 $1 $10 $100 $1,000 $10,000 $100,000 $1,000,000 $10,000,000 $100,000,00 0 Production $1,000,000, 000 Mean(VaR EVT / VaR Multi-Factor Model ) = 0.82 Source: Citigroup Page 29

30 Backtesting on Various Financial Assets HS L Proportion of Exceptions (1% in theory) N EVT-Hill EVT-MLE EVT-PWM SP % 1.12% 2.05% 1.12% 1.11% 1.11% Oil 1.01% 0.82% 1.52% 1.01% 1.01% 1.01% BRL 1.87% 1.25% 3.99% 1.50% 1.43% 1.43% KRW 1.92% 1.82% 2.79% 1.88% 1.90% 1.94% Gold 0.82% 0.78% 1.46% 0.89% 0.78% 0.82% Silver 0.87% 0.77% 1.37% 0.96% 0.84% 0.89% Source: Citigroup Page 30

31 Mean % Loss over VaR upon Exception HS L N EVT-Hill EVT-MLE EVT-PWM SP Oil BRL KRW Gold Silver Source: Citigroup Page 31

32 ES Fit (should be 1) HS L N EVT-Hill EVT-MLE EVT-PWM SP Oil BRL KRW Gold Silver Page 32 Source: Citigroup

33 Loss Allocation

34 Loss Allocation between trading divisions If a Hedge Fund defaults and losses occur in excess of the margin amount, how do we allocate the losses across several trading desks? Same problem as for capital allocation Pro-rata allocation method: P n = i i i= 1 i Si Allocation( S ) = VaR( S ) / VaR( S ) i Loss Marginal VaR approach: n P = S i= 1 Allocation( S ( P, S )/VaR( P) Loss i i ) = MargVaR i Page 34

35 Loss Allocation between trading divisions (Con d) No clear approach: should we reward a desk which happens to mitigate the bank s risk, most likely by chance? Indeed two desks could have large and almost offsetting positions, with CVaR of say 100MM and -99MM, for a resulting VaR of 1MM: in this case, loss allocation would be extreme / arbitrary Because portfolio tail risk can t be hedged away, all parameters of the model are questionable: they are not implied from the market A possible idea is to allocate losses based on a (ex-ante) recalibrated model Page 35

36 Loss Allocation between trading divisions (Con d) Pragmatic approach: Considering the sub-portfolio (P*) of all the desks with negative PL, we perturb the VaR model by bumping up all the volatility parameters of all the risk factors by a single number, so that the resulting VaR for the sub-portfolio matches the original portfolio VaR plus the money owed to the counterparty: VaR(P*, sigma*) = VaR(P, sigma) + payables The allocation is obtained by taking the Component VaR in the perturbed model. By definition, the sum of the component VaRs is equal to the total VaR: VaR ( P*, sigma*) = CVaR( P*, sigma*) i i VaR( Note that the Component VaR is defined as: CVaR( P) = Any resulting gain to a trading desk will be reallocated to the losing desks according to the method above i w i P) w i Page 36

37 Loss Allocation between trading divisions (Con d) Suppose the VaR margin collected from a hedge fund is $100MM, and they default on a MTM call for $120MM, with $180MM of receivables and $60MM of payables Desks which owe money to the counterparty won t be affected by the loss of $20MM Desks with losses (say desk 1 with $100MM and desk 2 with $80MM) will be allocated $160MM (the VaR plus the payables), by bumping the volatility parameters so that VaR (desk 1 + desk 2, sigma*) = $160MM Case 1: the Component VaRs for desk 1 and desk 2 are respectively $95MM and $65MM, we will allocate a loss of $5MM to desk 1 and a loss of $15MM to desk 2 Case 2: the Component VaRs for desk 1 and desk 2 are respectively $110MM and $50MM, we will allocate a loss of $30MM to desk 2, and instead of giving $10MM to desk 1, reallocate that amount to desk 2, which will incur a total loss of $20MM Page 37

38 A few conclusions

39 A few conclusions Beyond risk owned by the bank, it is important to understand the overall risk profile of the hedge fund even if only NAV returns are available Main model proposed here performed well, even in the current crisis EVT approach is limited: not a factor approach and therefore highly relying on historical data Heavy tailed multivariate distributions (such as T-distribution) are very difficult to handle in a high dimension space A more practical approach would be a regime switching model / mixture of normals: yet, it is not easy to define a stable and parsimonious set of regimes across thousand of market factors Page 39

40 Disclaimer This communication is issued by a member of the sales and trading department of Citigroup Global Markets Inc. or one of its affiliates (collectively, Citi ). Sales and trading department personnel are not research analysts, and the information in this communication ( Communication ) is not intended to constitute research as that term is defined by applicable regulations. Unless otherwise indicated, any reference to a research report or research recommendation is not intended to represent the whole report and is not in itself considered a recommendation or research report. All views, opinions and estimates expressed in this Communication (i) may change without notice and (ii) may differ from those views, opinions and estimates held or expressed by Citi or other Citi personnel. This Communication is provided for information and discussion purposes only. Unless otherwise indicated, it does not constitute an offer or solicitation to purchase or sell any financial instruments or other products and is not intended as an official confirmation of any transaction. Unless otherwise expressly indicated, this Communication does not take into account the investment objectives or financial situation of any particular person. Recipients of this Communication should obtain advice based on their own individual circumstances from their own tax, financial, legal and other advisors before making an investment decision, and only make such decisions on the basis of the investor's own objectives, experience and resources. The information contained in this Communication is based on generally available information and, although obtained from sources believed by Citi to be reliable, its accuracy and completeness cannot be assured, and such information may be incomplete or condensed. Citi often acts as an issuer of financial instruments and other products, acts as a market maker and trades as principal in many different financial instruments and other products, and can be expected to perform or seek to perform investment banking and other services for the issuer of such financial instruments or other products. The author of this Communication may have discussed the information contained therein with others within or outside Citi and the author and/or such other Citi personnel may have already acted on the basis of this information (including by trading for Citi's proprietary accounts or communicating the information contained herein to other customers of Citi). Citi, Citi's personnel (including those with whom the author may have consulted in the preparation of this communication), and other customers of Citi may be long or short the financial instruments or other products referred to in this Communication, may have acquired such positions at prices and market conditions that are no longer available, and may have interests different from or adverse to your interests. Page 40

41 Disclaimer (Con d) Investments in financial instruments or other products carry significant risk, including the possible loss of the principal amount invested. Financial instruments or other products denominated in a foreign currency are subject to exchange rate fluctuations, which may have an adverse effect on the price or value of an investment in such products. No liability is accepted by Citi for any loss (whether direct, indirect or consequential) that may arise from any use of the information contained in or derived from this Communication. Past performance is not a guarantee or indication of future results. Any prices provided in this Communication (other than those that are identified as being historical) are indicative only and do not represent firm quotes as to either price or size. You should contact your local representative directly if you are interested in buying or selling any financial instrument or other product or pursuing any trading strategy that may be mentioned in this Communication. Although Citibank, N.A. (together with its subsidiaries and branches worldwide, "Citibank") is an affiliate of Citi, you should be aware that none of the financial instruments or other products mentioned in this Communication (unless expressly stated otherwise) are (i) insured by the Federal Deposit Insurance Corporation or any other governmental authority, or (ii) deposits or other obligations of, or guaranteed by, Citibank or any other insured depository institution. IRS Circular 230 Disclosure: Citi and its employees are not in the business of providing, and do not provide, tax or legal advice to any taxpayer outside of Citi. Any statements in this Communication to tax matters were not intended or written to be used, and cannot be used or relied upon, by any taxpayer for the purpose of avoiding tax penalties. Any such taxpayer should seek advice based on the taxpayer s particular circumstances from an independent tax advisor Citigroup Global Markets Inc. Member SIPC. All rights reserved. Citi and Citi and Arc Design are trademarks and service marks of Citigroup Inc. or its affiliates and are used and registered throughout the world. Page 41