Framework for Assessing Market Risk

Transcription

1 Framework for Assessing Market Risk VAR Philippe Jorion University of California at Irvine May P.Jorion Please do not reproduce without author s permission Risk Management Methods: PLAN Components of risk measurement systems Value at Risk as a measure of downside risk Choice of VAR parameters: horizon and confidence level VAR caveats and alternative risk measures Stress tests The Basel Internal Model Approach Philippe Jorion 1

2 Risk Management (1) Components of risk measurement systems What is Market Risk? Market risk is the risk of losses from movements in the level or volatility of market prices, such as interest rates, foreign currencies, equities, and commodities Market risk measurement systems attempt to quantify the risk of losses in the market value (whether realized or unrealized) of the total portfolio The ultimate goal is to manage risks better Philippe Jorion 2

3 Components of a Risk Measurement System Positions Cash instrument #1 Sensitivity Notional Risk Factors Risk factor #1 Distribution Correlations Derivative instrument #1 Sensitivity Notional Risk factor #2 Distribution Market risk Evolution of Analytical Risk Management Tools 1938 Bond duration 1952 Markowitz mean-variance framework 1963 Sharpe's single factor model, systematic risk 1966 Multiple factor models 1973 Black-Scholes option pricing model, Greeks 1986 Limits on exposure by duration bucket 1988 Limits on exposure by Greeks 1993 Value at Risk 1997 VAR methods for credit risk Integration of credit and market risk Philippe Jorion 3

4 Evolution of Risk Management Systems (1) Limits on notionals» however, non-comparability across positions and losses unrelated to notional due to leverage Limits on sensitivities» however, not useful at institution s level; differences in volatilities across risk factors, correlations not taken into account Stop-loss limits» however, ex post Evolution of Risk Management Systems (2) Value at Risk (VAR) is a forward-looking measure of downside risk for the whole institution» takes into account current positions, leverage and diversification» allows comparisons across traders Limits on VAR and stress-test results» ex ante limits Philippe Jorion 4

5 Principles of Market Risk Measurement Objective: Obtain a good estimate of portfolio risk at a reasonable cost Steps: (1) choose a set of elementary risk factors and estimate their probability distribution (2) mapping : decompose financial instruments into exposures on these risk factors (3) aggregate the exposure for all positions and build the distribution of P&L on portfolio Constructing a Risk Measurement System Positions Global Repository Trades from front office Data feed with current prices Risk Factors Historical Market Data Mapping Risk Engine3a Model Positions Data Warehouse Portfolio Distribution Value at Risk Reports Distribution of Risk Factors Philippe Jorion 5

6 Tradeoffs in Risk Measurement Systems Modern risk measurement systems deal with very large portfolios Risk measurement uses tools from derivatives pricing but with some differences The risk manager must make simplifications, choosing risk factors that capture most risks» accuracy is not so important as in pricing, because VAR involves differences in values; also, errors wash out in large portfolios» speed of computation may be more important Outcome of Risk Measurement Systems Measure the downside risk of the value of a position W based on: (1) current position, assumed fixed over horizon (2) best estimate of risk environment Ideally, report the entire probability density function f(w) In practice, summarize by one number Philippe Jorion 6

7 Risk Management (2) Value at Risk as a measure of downside risk VALUE-AT-RISK VAR is a forward-looking method to express financial market risk in a form that anybody can understand--dollars Formally, VAR measures the predicted worst loss over a target horizon within a given confidence level» VAR is a measure of downside risk» VAR accounts for leverage and diversification effects and is more appropriate than notionals» VAR involves the art of the approximation Philippe Jorion 7

8 VAR: Definition VAR is the maximum loss over a target horizon such that there is a low, prespecified probability that the actual loss will be larger VAR(mean)= E(W)-W* 1 c = W* f ( w) dw= P( w W*) VAR is measured by the distribution quantile VAR can be measured relative to zero or to the mean, or discounted into the present Steps in the Computation of VAR Mark position to market Measure variability of risk factors Set time horizon Set confidence level Report potential loss Value Value Frequency Value σ VAR Sample computation: 10 days α Time Horizon Horizon $100M 15% (10/252) 2.33 = $7M Change to steps.swf Philippe Jorion 8

9 How to Measure VAR Define VAR as the worst dollar loss:» over a given horizon (T)» a confidence level (c, e.g. 95%)» the choice of these quantitative parameters depend on the nature of portfolio and use of VAR Simulate returns on the current portfolio using historical market data» map portfolio positions on selected risk factors» assume historical distribution relevant for future returns Computing VAR (1) Non-parametric approach: measure VAR from the sample quantile VAR(mean)= E(W)-W* (2) Parametric approach: assume/fit a distribution and measure VAR from sample standard deviation VAR(mean)= α σ(w) where α is the z-deviate that corresponds to confidence level (e.g for normal pdf) Philippe Jorion 9

10 How to Compute VAR? An Example (1) Consider the position of $4 billion short the yen, long the dollar: define Q 0 =$4 billion To assess potential moves in the spot rate, we take for instance ten years of historical data and assume that movements over the next day can be represented by historical changes Step 1: record 10 years of spot rate S t (yen/$) How to Compute VAR? An Example (2) Step 2: simulate the daily gain or loss on the position over the last ten year using R t ($) = Q0 ($)[ St St 1] / St 1 For instance, S 1 =112.0 and S 2 =111.8, which gives R 2 = $4,000m [ ]/112.0=-$7.2m Repeat over all days in the sample We have T= 2527 data points Philippe Jorion 10

11 How to Compute VAR? An Example (3) Simulated Daily Returns Return ($ million) $150 $100 $50 $0 -$50 -$100 -$150 1/2/90 1/2/91 1/2/92 1/2/93 1/2/94 1/2/95 1/2/96 1/2/97 1/2/98 1/2/99 How to Compute VAR? An Example (4) Construct a frequency distribution of losses Start ordering losses and count how many fall within ranges» below -$160m, we find 4 occurrences» between -$160m and -$140m, no losses» between -$140m and -$130m, 3 losses» and so on Plot the histogram of total number of losses against each range Philippe Jorion 11

12 How to Compute VAR? An Example (5) Distribution of Daily Returns Frequency % of observations VAR $160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Return ($ million) Change to VARhist.swf How to Compute VAR? An Example (6) We use a 95%=c confidence level We summarize the spread of the distribution by the 95% quantile, with p=100-95%=5% of the data in the left tail Here, the average gain or loss is close to zero We need to find the cutoff point R* such that p T = = 126 observations in left tail This gives VAR = $47.1m The maximum loss over one day is about $47 million at the 95 percent confidence level Philippe Jorion 12

13 Risk Management (3) Choice of VAR parameters: horizon and confidence level Choice of Quantitative Factors: Uses for VAR (1) Benchmark measure: to provide a companywide, time-consistent yardstick for risk» also, use multiplicative factor for capital adequacy (2) Potential loss measure: to give a broad idea of worst loss over horizon» liquidation period, time to hedge, period over which portfolio is fixed (3) Equity capital: to decide on the capital cushion to cover against market risk (4) Backtesting: to improve risk forecasting Philippe Jorion 13

14 Choice of Quantitative Factors (1) (1) Benchmark measure: confidence level and horizon arbitrary, but must be consistent across firm(s) and time (2) Potential loss measure:» horizon should reflect time needed for orderly portfolio liquidation for liquid bank portfolios (FX, GB), one day for illiquid securities, horizon must be longer regulators have chosen a 10-day horizon, sufficient for regulator to take over bank» confidence level arbitrary (reflects comfort level) Choice of Quantitative Factors (2) (3) Equity capital:» confidence level should be high enough to provide low probability of bankruptcy» horizon should be long enough to cover time required for corrective action--e.g.recapitalization (4) Backtesting:» confidence level should not be set too high, otherwise backtesting framework not powerful» horizon should be short (1-day) to have many independent observations, which improves power of tests Philippe Jorion 14

15 VAR as Equity Capital One-Year Default Rates Rating Frequency (Moody's) of Default Aaa 0.01% Aa3 0.03% A3 0.07% Baa3 0.70% Ba3 3.96% B1 6.14% B2 8.31% B % Measuring VAR: Effect of Parameters Horizon: volatility increases with square root of time, assuming (1) returns are not autocorrelated across days (2) the initial position is unchanged (no options) R 12 = R 1 + R 2, σ 2 (R 12 )= σ 2 (R 1 ) + σ 2 (R 2 ) +2 cov(r 1,R 2 ) σ(r T )= T σ(r 1 ) Confidence level: easy to transform VAR assuming normal distribution»e.g. c=95%, α=1.65 Philippe Jorion 15

16 VAR Parameters VARparameters.swf Measuring VAR: Changing the Parameters Example: transform VAR from RiskMetrics into VAR for Basle Committee» VAR RM = 95% over 1 day (α=1.65)» VAR BC = 99% over 10 days (α=2.33) Transform:» VAR BC = VAR RM (2.33/1.65) sqrt(10)» VAR BC = VAR RM (4.45) Philippe Jorion 16

17 Risk Management (4) VAR caveats-- Alternative measures of risk VAR Measures: Caveats (1) Frequency VAR does not describe the worst loss Empirical Histogram with VAR VAR» we would expect VAR to be exceeded with a frequency of p, or 5 days out of 100» the absolute worst loss in this sample is $214m» so, VAR does not give absolute worst loss $160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Profit/Loss ($ million) Change to VARworse.swf Philippe Jorion 17

18 Frequency VAR Measures: Caveats (2) VAR does not describe the losses in the left tail Histogram with Same VAR VAR -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Profit/Loss ($ million)» for the same VAR number, we could have very different distribution shapes» here, the average value of the losses worse than $47m is around $74m, which is 60% worse than VAR» we could keep VAR=-$47m and move (nearly) all losses below VAR to below -$160m Change to VARsame.swf Frequency VAR Measures: Caveats (3) VAR is measured with some error Histogram with Errors in VAR VAR 0 -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Profit/Loss ($ million)» VAR is subject to sampling variation (another number would have been found with another data sample)» there is no point in saying that VAR is $47,488,421» instead, we should say that VAR is around $47 million» VAR numbers are just broad estimates of downside risk Change to VARerror.swf Philippe Jorion 18

19 Alternative Measures of Risk (1) (1) Report the entire profit and loss distribution: The risk manager could report various quantiles at different confidence levels In theory, this is the best approach, as it reveals the extent of large losses In practice, the drawback of this approach is that it provides too much data Alternative Measures of Risk (2a) (2) Report the expected tail loss (ETL): This is defined as the expected value of the loss when it exceeds VAR (also called expected shortfall, conditional VAR, or expected tail loss) In theory, this is a better measure, especially for portfolios with options In practice, ETL measures may be imprecise if there are only a few observations in the left tail; instead, tail losses are typically estimated with stress tests Philippe Jorion 19

20 Alternative Measures of Risk (2b) Frequency Histogram with Expected Tail Loss VAR ETL -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Profit/Loss ($ million) The expected tail loss (ETL) is defined as N 1 ETL [ X < VAR] = ( x i ) N i= 1 This is the expected loss integrated over the tail area (N=126 observations) For example, for our yen position, this value is ETL = $74 million Change to VARETL.swf Alternative Measures of Risk (3a) (3) Report the standard deviation: For example, for our yen position, this is SD=$29.7 million In theory, this uses all of data points, not only those around the quantile, so is measured more precisely; also, it is sensitive to outliers, so should be able to highlight positions with large losses In practice, however, this measure, is symmetrical and treats gains and losses equally this may be acceptable for some positions but not for those with options Philippe Jorion 20

21 Alternative Measures of Risk (3b) With discrete data, the standard deviation (σ) is T 1 2 σ ( X ) = [ xi E( x)] ( T 1) i= 1» for example, assume that the profits and losses have a normal density function SD=$29.7 million» the normal deviate a at the 95% 1-tailed confidence level is 1.645; VAR is then αsd Sigma-based VAR= $49m» not very different from the historical VAR of $47m Risk Management (5) Stress Tests Philippe Jorion 21

22 Why Stress-Testing? VAR does not measures the absolute worst loss that could happen; the risk management system may have other flaws VAR measures must be complemented by stress-testing, which aims at identifying situations that could create extraordinary losses for the institution Stress-testing is required by the Basel Committee as a precondition for using internal VAR models Stress Tests: Why not VAR? In theory, increasing the VAR confidence level could uncover large losses In practice, stress tests attempt to discover scenarios that would not occur under standard VAR methods (1) Simulating shocks that never occurred, or did not occur with sufficient frequency (e.g. in recent historical data) (2) Simulating shocks that reflect structural breaks (e.g. devaluations) Philippe Jorion 22

23 What is Stress-Testing? Stress-testing is a key risk management process, which includes (i) scenario analysis, (ii) stressing models, volatilities and correlations, and (iii) developing policy responses to stress tests Scenario analysis submits the portfolio to large movements in financial market variables The objective of stress-testing and management response should be to ensure that the institution can withstand likely scenarios without going bankrupt Scenario Analysis: Univariate Scenarios (1) Moving key variables one at a time:» simple and intuitive method»example: the portfolio is long the dollar vs. yen we suppose the dollar could fall by 15% in one week; this gives a worst loss of $600 million» problem is with multiple sources of risk: if the portfolio also contains positions in Japanese and US equities, we would have to predict movements in these markets as well we cannot assume the worst loss will occur at the same time in all markets Philippe Jorion 23

24 Scenario Analysis: Historical Scenarios (2) Historical scenarios» automatically account for correlations» typical choices: 1987 stock market crash, devaluation of the British pound in 1992, bond market debacle of 1984»example: the portfolio has positions of $4b long dollar/yen, plus $4b long U.S. equities and $4b short Japanese equities during the week of October 2, 1998, the dollar fell by 13.9%, S&P by 1.8% and Nikkei by 2.6%: the total loss would have been $732 million Scenario Analysis: Prospective Scenarios (3) Creating prospective scenarios» useful when the past offers little guidance for extreme movements» for instance, the portfolio may be exposed to a fixed exchange rate; this does not mean that there is no economic risk, since a devaluation could occur» ideally, the scenario should be tailored to the portfolio at hand, assessing the worst thing that could happen Philippe Jorion 24

25 Stress Tests: Problems Scenarios inherently subjective Scenarios should be driven by the risk exposures of current portfolio Problem is to generalize from movements in a few risk factors to total portfolio risk It is difficult to attach probabilities to scenarios extreme events Results of scenarios may involve catastrophic losses and are often ignored Risk Management (6) The Basel Internal Model Approach Philippe Jorion 25

26 CAPITAL ADEQUACY: Basel Market Rules The computation of VAR shall be based on a set of uniform quantitative inputs:» a horizon of 10 trading days, or two calendar weeks (T)» a 99 percent confidence interval (c)» an observation period based on at least a year of historical data and updated at least once a quarter Market Risk Charge is set at the higher of:» the previous day's VAR, and» the average VAR over the last sixty business days, times a multiplier, k: MRC(t) = Max[ k (1/60)Σ i=1 60 VAR(t-i), VAR(t-1)]+SRC Internal Models: Qualitative Criteria Internal model can only be used when: (a) banks have an independent risk control unit (b) bank conducts back-testing (c) board/senior management is involved (d) internal model is used to monitor risk (e) trading and exposure limits also exist (f) stress testing is also used (g) documentation for compliance exists (h) independent reviews are done regularly Philippe Jorion 26

27 Internal Model: The Multiplier Multiplier: the value of k is determined by local regulators, subject to a floor of three:» k is intended to provide additional protection against unusual environments (otherwise, 1 failure very 4 years) Plus factor: a penalty component shall be added to k if back-testing reveals that the bank's internal model incorrectly forecasts risks, or internal risk management practices are viewed as inadequate Why the Multiplicative Factor? To protect against model risk, or fat tails For any random variable x with finite variance, Chebyshev s inequality states»p[ x-µ >rσ] 1/r 2» if symmetric, P[(x-µ)<-rσ] (1/2)1/r 2» set RHS to 1%; then r=7.071, VAR MAX =7.071σ» with a normal distribution VAR N =2.326σ» correction k= VAR MAX /VAR N =3.03 There is arbitrariness in the joint choice of (c, T and k) Philippe Jorion 27

28 Internal Model: Equivalent Risk Charges Horizon: Confidence 99.99% Aaa 99.9% A3 99% Baa3 97.5% Ba3 95% B1 90% B2 84.1%(1xσ) B3 Normal and independent distribution VAR Reporting: 1998 Institution Confidence Level Average 1-day VAR ($MM) 99%VAR ($MM) MR Charge ($MM) Risk Capital ($MM) Bank America 97.5% ,055 Bankers Trust 99% ,540 Citicorp 97.7% ,008 Chase 99% ,161 J.P. Morgan 95% ,454 Deutsche Bank 99% ,303 UBS 99% ,322 Barclays 98% ,953 Bear Stearns 95% ,955* Merrill Lynch 99% ,132* Morgan Stanley 99% ,119* Salomon 99% ,768* CSFP 99% ,257* Source: Bank financial reports. * refers to equity capital only. Philippe Jorion 28

29 Risk Management Conclusions CONCLUSIONS (1) Market risk measurement attempts to predict the distribution of losses on a portfolio Downside risk can be summarized with a single measure, VAR, defined at a given confidence level over a certain horizon VAR should be complemented by stress tests, based on scenario analysis Philippe Jorion 29

30 CONCLUSIONS (2) Models are usually based on historical information that may not reflect future risks Models involve simplifications; risk manager must understand whether risk model captures risk of strategy Models assume current positions are frozen over the horizon, and ignore liquidity issues The ultimate goal of risk measurement is to understand risk better so as to manage it effectively References Philippe Jorion is Professor of Finance at the Graduate School of Management at the University of California at Irvine He is the author of Value at Risk: The New Benchmark for Managing Financial Risk published by McGrawHill (2001) and the Financial Risk Manager Handbook published by Wiley (2003) He is also editor of the Journal of Risk, published by Risk Publications Phone: (949) FAX: (949) Web: Philippe Jorion 30

31 It Pays to Learn! Market Risk Management Series A New Series of Online Courses for Managing Market Risk The Market Risk Management Series of five courses provides a complete understanding of market risk and the issues involved in measuring, computing and managing that risk. The courses cover the concepts tools and methods of market risk management including identifying sources of risk, derivatives, hedging and VAR methods. Who can Benefit This series is designed using a comprehensive and progressive approach and is therefore suitable for practitioners at all levels of experience who manage exposure to market risk. These include: Treasurers, assistant treasurers and other risk officers who manage market risk exposures in commercial firms, commercial banks, central banks, investment banks, mutual funds, pension funds, brokerage firms and insurances companies Financial analysts Executives whose firms are exposed to market risk Professionals who need to understand how market risk is measured, assessed and managed in a diversified portfolio of financial instruments (equity, fixed-income, currency, commodity). Individuals preparing for the FRM TM exam will also find the course useful for exam preparation. The courses include sample questions from recent FRM TM exams Developer Developed by Dr. Philippe Jorion (University of California at Irvine), the courses incorporate a sound pedagogical approach that emphasizes practical application and developing strong analytical and problem solving skills. State-of-the-art Web Delivery Each course in the series was designed for the Web and uses the latest in multimedia design to deliver an exceptional learning experience to participants. The courses feature many detailed examples, narration, interactive equations, and self-assessment questions. For Enrollment and Other Information visit or call

32 Market Risk Management Series Topics Market Risk Management Series Outline of the five courses Course 1 Introduction to Market Risk (code 411) Definition Review of Probability Distributions Measuring Value at Risk (VAR) Choosing VAR Parameters Pros and Cons of VAR Measures Stress-Testing Course 2 Sources of Market Risk (code 412) Types of Risk Sources of Market Risk Fixed-Income Risk Equity Risk Currency Risk Commodity Risk Liquidity Risk Course 3 Managing Linear Risk (code 421) Pricing of Forward and Futures Contracts Risk Management with Forwards and Futures Optimal Hedging Hedging Fixed-Income, Equity, and Option Portfolios Course 4 Managing Non-Linear Risk (code 422) Features of Option Contracts Pricing of Options Mapping Options on Risk Factors Measuring Exposures for Complex Portfolios Risk Management with Options Course 5 VAR Methods (code 423) Introduction to Modern Risk Measurement Systems Local versus Full Valuation Methods Mapping Positions on Risk Factors Modeling Portfolio Risk VAR Measurement Methods About The Derivatives Institute The Derivatives Institute is the training service of the Montréal Exchange Canada s only financial derivatives exchange. As Canada s oldest exchange, the Montréal Exchange has a long and distinguished history of offering training to the financial community. In 2001, the Exchange further expanded its training service by creating The Derivatives Institute as a key component of the Exchange s reorientation as a specialized derivatives exchange, with the goal of providing practical and applied derivatives education training worldwide.