Assessing Market Risk
Framework for Assessing Market Risk
VAR
Philippe Jorion University of California at Irvine July 2004
Please do not reproduce © 2004 P.Jorion without author’s permission E-mail: [email protected]
Framework for Assessing Market Risk
411-ecs60931.swf; VARstart.swf
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Assessing Market Risk: PLAN
(1) Components of risk measurement systems (2) Value at Risk as a measure of downside risk (3) Choice of VAR parameters: horizon and confidence level (4) VAR caveats and alternative risk measures (5) Stress tests
Risk Management - Philippe Jorion
Assessing Market Risk
(1) Components of risk measurement systems
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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
Risk Management - Philippe Jorion
Components of a Risk Measurement System
Positions Risk Factors Cash instrument #1 Risk factor #1 Sensitivity Notional Distribution
Correlations
Derivative instrument #1 Risk factor #2 Sensitivity Distribution Notional
Market risk
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Evolution of Analytical Risk Management Tools
Risk Management - Philippe Jorion 411-ecs13504.swf; Evolution.swf
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
1998- Integration of credit and market risk
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Evolution of Market 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
Risk Management - Philippe Jorion
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
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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
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Constructing a Risk Measurement System
Risk Management - Philippe Jorion 424-ecs41441.swf; System.swf
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Constructing a Risk Measurement System
Positions Risk Factors Trades from front office Global Historical Repository Data feed with Market Data current prices
Mapping Model
Risk Engine3a
Portfolio Distribution Distribution Positions of Risk Factors Value at Risk Data Warehouse Reports
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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
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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
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Example: JP Morgan Chase 2003 Annual Report (1) Tools used to measure risks: ... the Firm uses several measures, both statistical and nonstatistical, including: ! Statistical risk measures: » Value-at-Risk (“VAR”) ! Nonstatistical risk measures: » Stress tests » Measures of position size and sensitivity
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Example: JP Morgan Chase 2003 Annual Report (2)
Value-at-Risk JPMorgan Chase’s statistical risk measure, VAR, gauges the potential loss from adverse market moves in an ordinary market environment and provides a consistent cross-business measure of risk profiles and levels of risk diversification. VAR is used to compare risks across businesses, to monitor limits and to allocate economic capital to the business segments. VAR provides risk transparency in a normal trading environment.
Risk Management - Philippe Jorion
Example: JP Morgan Chase 2003 Annual Report (3)
Value-at-Risk Each business day, the Firm undertakes a comprehensive VAR calculation that includes both trading and nontrading activities. JPMorgan Chase’s VAR calculation is highly granular, comprising more than 1.5 million positions and 240,000 pricing series (e.g., securities prices, interest rates, foreign exchange rates). For a substantial portion of its exposure, the Firm has implemented full-revaluation VAR, which, management believes, generates the most accurate results.
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Example: JP Morgan Chase 2003 Annual Report (4) Value-at-Risk To calculate VAR, the Firm uses historical simulation, which measures risk across instruments and portfolios in a consistent, comparable way. This approach assumes that historical changes in market value are representative of future changes. The simulation is based on market data for the previous 12 months. The Firm calculates VAR using a one-day time horizon and a 99% confidence level. This means the Firm would expect to incur losses greater than that predicted by VAR estimates only once in every 100 trading days, or about 2.5 times a year. Risk Management - Philippe Jorion
Assessing Market Risk
(2) Value at Risk as a measure of downside risk
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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”
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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* W* 1−c = 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
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Steps in the Computation of VAR
Risk Management - Philippe Jorion 411-ecs18071.swf; Steps.swf
Steps in the Computation of VAR
Mark Measure Set Set Report position variability of time confidence potential to market risk factors horizon level loss
Value Value Frequency Value
VAR σ
10 days
Time Horizon−α Horizon Sample computation: $100M × 15% ×√(10/252) × 2.33 = $7M
Change to steps.swf Risk Management - Philippe Jorion
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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
Risk Management - Philippe Jorion
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. 1.65 for normal pdf)
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How to Compute VAR? An Example (1)
! Consider the position of $4 billion short the yen, long the dollar: define Q0=$4 billion ! To assess potential moves in the spot rate, we take for instance ten years of historical data and assume that movements over the next day can be represented by historical changes Step 1: record 10 years of spot rate
St(yen/$)
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How to Compute VAR? An Example (2)
Step 2: simulate the daily gain or loss on the position over the last ten year using
Rt ($)= Q0 ($)[St − St−1]/ St−1
! For instance, S1=112.0 and S2=111.8, which gives
R2= $4,000m × [111.8-112.0]/112.0=-$7.2m ! Repeat over all days in the sample
! We have T= 2527 data points
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How to Compute VAR? An Example (3)
Simulated Daily Returns Return ($ 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
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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
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How to Compute VAR? An Example (5)
Distribution of Daily Returns
Frequency 400
350 VAR 300 5% of observations 250
200
150
100
50
0 -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Return ($ million) Risk Management - Philippe Jorion 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 = 0.05 × 2527 = 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”
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Example: JP Morgan Chase 2003 Annual Report (5) Value-at-Risk: average $69 million
$69m
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Example: JP Morgan Chase 2003 Annual Report (6) Value-at-Risk
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Assessing Market Risk
(3) Choice of VAR parameters: horizon and confidence level
Choice of Quantitative Factors: Uses for VAR
(1) Benchmark measure: to provide a company- wide, 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
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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) Risk Management - Philippe Jorion
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 Risk Management - Philippe Jorion
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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% B3 15.08% Risk Management - Philippe Jorion
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)
R12 = R1+ R2, 2 2 2 σ (R12)= σ (R1) + σ (R2) +2 cov(R1,R2)
σ(RT)= √T σ(R1) ! Confidence level: easy to transform VAR assuming normal distribution »e.g. c=95%, α=1.65
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VAR Parameters
VARparameters.swf
Measuring VAR: Changing the Parameters
! Example: transform VAR from RiskMetrics into VAR for Basle Committee
» VARRM = 95% over 1 day (α=1.65)
» VARBC = 99% over 10 days (α=2.33) ! Transform:
» VARBC = VARRM (2.33/1.65) sqrt(10)
» VARBC = VARRM (4.45) ! This assumes independent identical normal distributions
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JP Morgan: Daily VAR, 1994-98
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Assessing Market Risk
(4) VAR caveats: Alternative measures of risk
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VAR does not describe the worst loss
Risk Management - Philippe Jorion 411-ecs18135.swf; VARworse.swf
VAR Measures: Caveats (1) VAR does not describe the worst loss
Empirical Histogram with VAR Frequency » we would expect VAR to be 500 exceeded with a frequency of 450 VAR
400 p, or 5 days out of 100 350 » the absolute worst loss in this 300
250 sample is $214m 200 » so, VAR does not give 150
100 absolute worst loss
50
0 -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 Profit/Loss ($ million)
Risk Management - Philippe Jorion Change to VARworse.swf
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VAR does not describe the losses in the left tail
Risk Management - Philippe Jorion 411-ecs13540.swf; VARsame.swf
VAR Measures: Caveats (2) VAR does not describe the losses in the left tail Histogram with Same VAR » for the same VAR number, Frequency 500 we could have very different 450 VAR
400 distribution shapes 350 » here, the average value of 300
250 the losses worse than $47m 200 is around $74m, which is 150 60% worse than VAR 100 50 » we could keep VAR=-$47m 0 -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 and move (nearly) all losses Profit/Loss ($ million) below VAR to below -$160m
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VAR is measured with some error
Risk Management - Philippe Jorion 411-ecs18141.swf; VARerror.swf
VAR Measures: Caveats (3) VAR is measured with some error
Histogram with Errors in VAR » VAR is subject to sampling Frequency 500 variation (another number 450 VAR
400 would have been found with 350 another data sample) 300
250 » there is no point in saying 200 that VAR is $47,488,421 150
100 » instead, we should say that 50 VAR is around $47 million 0 -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 » VAR numbers are just broad Profit/Loss ($ million) estimates of downside risk
Risk Management - Philippe Jorion Change to VARerror.swf
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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
Risk Management - Philippe Jorion
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
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VAR and Expected Tail Loss (ETL)
Risk Management - Philippe Jorion 411-ecs18150.swf; VARETL.swf
Alternative Measures of Risk (2b)
! The expected tail loss Histogram with Expected Tail Loss (ETL) is defined as Frequency 500 N 450 VAR 1 ETL [X < −VAR] = ( x ) 400 ∑ i ETL N 350 i=1 300 ! 250 This is the expected loss 200 integrated over the tail 150 area (N=126 observations) 100 50 ! For example, for our yen 0 -$160 -$120 -$80 -$40 $0 $40 $80 $120 $160 position, this value is Profit/Loss ($ million) ETL = $74 million Risk Management - Philippe Jorion Change to VARETL.swf
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VAR and Expected Tail Loss Normal distribution Confidence 99.99 99.9 99 95 90 50 Quantile -3.715 -3.090 -2.326 -1.645 -1.282 0.000 Tail loss -4.018 -3.370 -2.667 -2.062 -1.754 -0.798
! Tail loss close to the quantile due to the fast dropoff in tails—not necessarily the case with other distributions
Value at Risk - P.Jorion
COHERENT RISK MEASURES: Artzner et al. (1999)
! Desirable properties for risk measures ρ(W) (1) Monotonicity: if W1≤ W2, ρ(W1)≥ρ(W2) (if a portfolio has lower returns for all states of the world, its risk must be greater) (2) Translation Invariance: ρ(W+k) = ρ(W)-k (adding cash k to W should reduce its risk by k) (3) Homogeneity: ρ(bW)= bρ(W) (scaling a portfolio should simply scale its risk) (4) Subadditivity: ρ(W1+W2) ≤ρ(W1)+ρ(W2) (merging portfolios cannot increase risk)
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Coherent Risk Measures VAR and Expected Tail Loss
! Quantile-based VAR measure fails to satisfy the last property » pathological examples of short option positions can create large losses with a low probability and hence have low VAR, yet combine to create portfolios with larger VAR ! Shortfall measure E[-X| X<-VAR] satisfies the desirable “coherence” properties ! With elliptical distributions, however, the standard deviation-based VAR satisfies the last property Value at Risk - P.Jorion
Why VAR is not Necessarily Subadditive
! Consider an investment in a corporate bond with face value of $100,000 and default probability of 0.5%; the portfolio has 3 such bonds, with independent defaults ! For each bond, payoffs are -$100,000 with probability of 0.5% and +$500 with prob 99.5%; the 99% VAR is $500 State Probability Payoff No default 0.9953 =0.985075 $1,500 1 default 3*0.005*0.9952 =0.014850 -$99,000 2 defaults 3*0.0052 *0.9953 =0.000075 -$199,500 3 defaults 0.0053 =0.0000001 -$300,000 ! Sum of VARs=$1,500; portfolio VAR=$131,991 (interpolate) ! Thus ρ(ΣW) > Σ ρ(W): VAR is not subadditive
Value at Risk - P.Jorion
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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 Risk Management - Philippe Jorion
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
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Assessing Market Risk
(5) Stress Tests
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 Risk Management - Philippe Jorion
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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)
Risk Management - Philippe Jorion
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 Risk Management - Philippe Jorion
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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 Risk Management - Philippe Jorion
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
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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
Risk Management - Philippe Jorion
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
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Example: JP Morgan Chase 2003 Annual Report (7)
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Example: JP Morgan Chase 2003 Annual Report (8) Stress Tests The potential stress-test loss as of December 4, 2003, is the result of the “Equity Market Collapse” stress scenario, which is broadly modeled on the events of October 1987. Under this scenario, » global equity markets suffer a sharp reversal after a long sustained rally; equity prices decline globally; » volatilities for equities, interest rates and credit products increase dramatically for short maturities and less so for longer maturities; » sovereign bond yields decline moderately; and » swap spreads and credit spreads widen.
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Assessing Market Risk
(6) 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
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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 Risk Management - Philippe Jorion
References
! Philippe Jorion is Professor of Finance at the Graduate School of Management at the University of California at Irvine ! Author of “Value at Risk,” published by McGraw-Hill in 1997, which has become an “industry standard,” translated into 7 other languages; revised in 2000 ! Author of the “Financial Risk Manager Handbook,” published by Wiley and exclusive text for the FRM exam; revised in 2003 ! Editor of the “Journal of Risk” ! Some of this material is based on the online "market risk management" course developed by the Derivatives Institute: for more information, visit www.d-x.ca, or call 1-866-871-7888
Phone: (949) 824-5245 E-Mail: [email protected] FAX: (949) 824-8469 Web: www.gsm.uci.edu/~jorion
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