assessing market risk by jorion philippe
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Assessing Market Risk
Philippe Jorion 2
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 riskmeasurement systems
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What is Market Risk?! Market risk is the risk of losses from
movements in the level or volatility of marketprices, such as interest rates, foreigncurrencies, equities, and commodities
! Market risk measurement systems attempt toquantify the risk of losses in the market value(whether realized or unrealized) of the totalportfolio
! The ultimate goal is to manage risks better
Risk Management - Philippe Jorion
Distribution
Risk Factors
Marketrisk
Correlat ions
Distribution
Risk factor #1
Risk factor #2
Cash instrument #1
Positions
Derivative instrument #1Sensitivity
SensitivityNotional
Notional
Components of a
Risk Measurement System
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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
Risk Management - Philippe Jorion
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 positionsand 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 wholeinstitution» 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 andestimate their probability distribution
(2) “mapping”: decompose financial instrumentsinto exposures on these risk factors
(3) aggregate the exposure for all positions andbuild the distribution of P&L on portfolio
Risk Management - Philippe Jorion
Constructing a Risk Measurement System
Risk Management - Philippe Jorion 424-ecs41441.swf; System.swf
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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 overhorizon
(2) best estimate of risk environment
! Ideally, report the entire probability densityfunction f(W)
! In practice, summarize by one number
Risk Management - Philippe Jorion
Example: JP Morgan Chase
2003 Annual Report (1)Tools used to measure risks:
... the Firm uses several measures, bothstatistical 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 movesin an ordinary market environment and provides aconsistent cross-business measure of risk profilesand levels of risk diversification. VAR is used tocompare risks across businesses, to monitor limitsand to allocate economic capital to the business
segments. VAR provides risk transparency in anormal trading environment.
Risk Management - Philippe Jorion
Example: JP Morgan Chase
2003 Annual Report (3)Value-at-Risk
Each business day, the Firm undertakes acomprehensive VAR calculation that includes bothtrading and nontrading activities. JPMorgan Chase’sVAR calculation is highly granular, comprising morethan 1.5 million positions and 240,000 pricing series(e.g., securities prices, interest rates, foreign
exchange rates). For a substantial portion of itsexposure, the Firm has implemented full-revaluationVAR, which, management believes, generates themost 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 portfoliosin a consistent, comparable way. This approachassumes that historical changes in market value arerepresentative of future changes. The simulation isbased 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 Firmwould expect to incur losses greater than thatpredicted by VAR estimates only once in every 100trading days, or about 2.5 times a year.
Risk Management - Philippe Jorion
Assessing Market Risk
(2)
Value at Risk as a measure ofdownside risk
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VALUE-AT-RISK! VAR is a forward-looking method to express
financial market risk in a form that anybodycan understand--dollars
! Formally, VAR measures the predicted“worst” loss over a target horizon within agiven confidence level» VAR is a measure of downside risk
» VAR accounts for leverage and diversificationeffects and is more appropriate than notionals
» VAR involves the “art of the approximation”Risk Management - Philippe Jorion
VAR: Definition! VAR is the maximum loss over a target
horizon such that there is a low, prespecifiedprobability that the actual loss will be larger
VAR(mean)= E(W)-W*
! VAR is measured by the distribution quantile
! VAR can be measured relative to zero or tothe mean, or discounted into the present
! "# $%%#
**)()(1
W W w P dww f c
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Steps in the Computation of VAR
Risk Management - Philippe Jorion 411-ecs18071.swf; Steps.swf
Markpositionto market
Settimehorizon
Setconfidencelevel
Value
Time Horizon Horizon
Value ValueFrequency
#&
Reportpotentialloss
Measurevariability of risk factors
$100M ' ((10/252) ' 2.33 = $7M' 15%
)
VAR
10 days
Sample computation:
Steps in the Computation of VAR
Risk Management - Philippe JorionChange to steps.swf
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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 parametersdepend on the nature of portfolio and use of VAR
! Simulate returns on the current portfoliousing 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 adistribution and measure VAR from samplestandard deviation
VAR(mean)= & )(W)where & is the z-deviate that corresponds toconfidence 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=$4 billion
! To assess potential moves in the spot rate, we takefor instance ten years of historical data and assumethat movements over the next day can berepresented by historical changes
Step 1: record 10 years of spot rate
S t (yen/$)
Risk Management - Philippe Jorion
How to Compute VAR?
An Example (2)Step 2: simulate the daily gain or loss on the position
over the last ten year using
! For instance, S 1=112.0 and S 2=111.8, which gives
R2= $4,000m ' [111.8-112.0]/112.0=-$7.2m
! Repeat over all days in the sample! We have T= 2527 data points
Risk Management - Philippe Jorion
110 /]($)[($) ###% t t t t S S S Q R
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How to Compute VAR?
An Example (3)Return ($ 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
Simulated Daily Returns
Risk Management - Philippe Jorion
How to Compute VAR?
An Example (4)! Construct a frequency distribution of losses
! Start ordering losses and count how many fallwithin 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 againsteach range
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How to Compute VAR?
An Example (5)
VAR 5% ofobservations
0
50
100
150
200
250
300
350
400
-$160 -$120 -$80 -$40 $0 $40 $80 $120 $160
Frequency
Return ($ million)
Distribution of Daily Returns
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 the95% quantile, with p=100-95%=5% of the data inthe 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 millionat the 95 percent confidence level”
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Example: JP Morgan Chase
2003 Annual Report (5)Value-at-Risk: average $69 million
Risk Management - Philippe Jorion
$69m
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 ideaof worst loss over horizon» liquidation period, time to hedge, period over
which portfolio is fixed
(3) Equity capital: to decide on the capitalcushion 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 consistentacross 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 toprovide low probability of bankruptcy
» horizon should be long enough to cover timerequired 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 manyindependent observations, which improves powerof tests
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VAR as Equity Capital
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
One-Year Default Rates
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(R12)= )2(R1) + )2(R2) +2 cov(R1,R2)
)(RT)= (T )(R
1)
! Confidence level: easy to transform VARassuming normal distribution» e.g. c =95%, &=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 (&=1.65)
» VARBC = 99% over 10 days (&=2.33)
! Transform:
» VARBC = VARRM (2.33/1.65) sqrt(10)
» VARBC = VARRM (4.45)
! This assumes independent identical normaldistributions
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JP Morgan:
Daily VAR, 1994-98
Risk Management - Philippe Jorion
Assessing Market Risk
(4)
VAR caveats:
Alternative measures of risk
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Risk Management - Philippe Jorion
VAR does not describe the worst loss
411-ecs18135.swf; VARworse.swf
VAR Measures: Caveats (1)
» we would expect VAR to beexceeded with a frequency of
p, or 5 days out of 100
» the absolute worst loss in thissample is $214m
» so, VAR does not give
absolute worst loss
Risk Management - Philippe Jorion
Empirical Histogram with VAR
0
50
100
150
200
250
300
350
400
450
500
-$160 - $120 -$80 -$40 $0 $40 $80 $120 $160
Frequency
Profit/Loss ($ million)
VAR
VAR does not describe the worst loss
Change to VARworse.swf
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Risk Management - Philippe Jorion
VAR does not describe the losses in the left tail
411-ecs13540.swf; VARsame.swf
VAR Measures: Caveats (2)
» for the same VAR number,we could have very differentdistribution shapes
» here, the average value ofthe losses worse than $47mis around $74m, which is
60% worse than VAR» we could keep VAR=-$47m
and move (nearly) all lossesbelow VAR to below -$160m
Risk Management - Philippe Jorion
Histogram with Same VAR
0
50
100
150
200
250
300
350
400
450
500
-$160 - $120 -$80 -$40 $0 $40 $80 $120 $160
Frequency
Profit/Loss ($ million)
VAR
VAR does not describe the losses in the left tail
Change to VARsame.swf
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Risk Management - Philippe Jorion
VAR is measured with some error
411-ecs18141.swf; VARerror.swf
VAR Measures: Caveats (3)
» VAR is subject to samplingvariation (another numberwould have been found withanother data sample)
» there is no point in sayingthat VAR is $47,488,421
» instead, we should say thatVAR is around $47 million
» VAR numbers are just broadestimates of downside risk
Risk Management - Philippe Jorion
VAR is measured with some error
Histogram with Errors in VAR
0
50
100
150
200
250
300
350
400
450
500
-$160 -$120 -$80 -$40 $0 $40 $80 $120 $160
Frequency
Profit/Loss ($ million)
VAR
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 atdifferent confidence levels
! In theory, this is the best approach, as it reveals theextent of large losses
! In practice, the drawback of this approach is that itprovides 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 losswhen it exceeds VAR (also called expectedshortfall, conditional VAR, or expected tail loss)
! In theory, this is a better measure, especially forportfolios with options
!
In practice, ETL measures may be imprecise ifthere are only a few observations in the left tail;instead, tail losses are typically estimated withstress tests
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Risk Management - Philippe Jorion
VAR and Expected Tail Loss (ETL)
411-ecs18150.swf; VARETL.swf
Alternative Measures of Risk (2b)! The expected tail loss
(ETL) is defined as
! This is the expected lossintegrated over the tail
area ( N =126 observations)! For example, for our yen
position, this value is
ETL = $74 millionRisk Management - Philippe Jorion
Histogram with Expected Tail Loss
0
50
100
150
200
250
300
350
400
450
500
-$160 - $120 -$80 -$40 $0 $40 $80 $120 $160
Frequency
Profit/Loss ($ million)
VAR
ETL
)1
(][
1
+%
%#, N
ii x
N VAR X ETL
Change to VARETL.swf
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VAR and Expected Tail Loss
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
Normal distribution
! Tail loss close to the quantile due to the fastdropoff in tails—not necessarily the casewith other distributions
Value at Risk - P.Jorion
COHERENT RISK MEASURES:
Artzner et al. (1999)! Desirable properties for risk measures -(W)(1) Monotonicity: if W1$ W2, -(W1).-(W2)
(if a portfolio has lower returns for all states of theworld, its risk must be greater)
(2) Translation Invariance: -(W+k) = -(W)-k(adding cash k to W should reduce its risk by k)
(3) Homogeneity: -(bW)= b-(W)(scaling a portfolio should simply scale its risk)(4) Subadditivity: -(W1+W2) $ -(W1)+-(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 probabilityand hence have low VAR, yet combine tocreate portfolios with larger VAR
! Shortfall measure E[-X| X / -(W): VAR is not subadditive
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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 thosearound the quantile, so is measured moreprecisely; also, it is sensitive to outliers, so shouldbe able to highlight positions with large losses
! In practice, however, this measure, is symmetricaland treats gains and losses equally—this may beacceptable for some positions but not for those withoptions
Risk Management - Philippe Jorion
Alternative Measures of Risk (3b)! With discrete data, the standard deviation ()) is
» for example, assume that the profits and losses have anormal density function SD=$29.7 million
» the normal deviate a at the 95% 1-tailed confidence levelis 1.645; VAR is then &SD
Sigma-based VAR= $49m
» not very different from the historical VAR of $47m
Risk Management - Philippe Jorion
+%
##
%T
i
i x E xT
X
1
2)]([)1(
1)()
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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 managementsystem may have other flaws
! VAR measures must be complemented bystress-testing, which aims at identifyingsituations that could create extraordinary
losses for the institution! Stress-testing is required by the Basel
Committee as a precondition for usinginternal VAR models
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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 discoverscenarios that would not occur understandard VAR methods
(1) Simulating shocks that never occurred, ordid 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 largemovements in financial market variables
! The objective of stress-testing and managementresponse should be to ensure that the institutioncan withstand likely scenarios without goingbankrupt
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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 inthese markets as well
– we cannot assume the worst loss will occur at thesame 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 losswould have been $732 million
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Scenario Analysis:
Prospective Scenarios(3) Creating prospective scenarios
» useful when the past offers little guidance forextreme movements
» for instance, the portfolio may be exposed to afixed exchange rate; this does not mean thatthere is no economic risk, since a devaluationcould occur
» ideally, the scenario should be tailored to theportfolio at hand, assessing the worst thing thatcould happen
Risk Management - Philippe Jorion
Stress Tests:
Problems! Scenarios inherently subjective
! Scenarios should be driven by the riskexposures of current portfolio
! Problem is to generalize from movements ina 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)
Risk Management - Philippe Jorion
Example: JP Morgan Chase
2003 Annual Report (8)Stress Tests
The potential stress-test loss as of December 4, 2003, isthe result of the “Equity Market Collapse” stressscenario, which is broadly modeled on the events ofOctober 1987. Under this scenario,
» global equity markets suffer a sharp reversal after along sustained rally; equity prices decline globally;
» volatilities for equities, interest rates and credit
products increase dramatically for short maturitiesand less so for longer maturities;
» sovereign bond yields decline moderately; and
» swap spreads and credit spreads widen.Risk Management - Philippe Jorion
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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 asingle measure, VAR, defined at a givenconfidence 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 managermust understand whether risk modelcaptures risk of strategy
! Models assume current positions are frozenover the horizon, and ignore liquidity issues
! The ultimate goal of risk measurement is tounderstand risk better so as to manage iteffectively
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 7other languages; revised in 2000
! Author of the “Financial Risk Manager Handbook,” publishedby Wiley and exclusive text for the FRM exam; revised in2003
! Editor of the “Journal of Risk”
! Some of this material is based on the online "market riskmanagement" course developed by the Derivatives Institute:
for more information, visit www.d-x.ca, or call 1-866-871-7888
Phone: (949) 824-5245
FAX: (949) 824-8469
E-Mail: [email protected]
Web: www.gsm.uci.edu/~jorion
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