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Measuring Market Risk
Philippe Jorion 1
Measuring Market Risk
Philippe JorionUniversity of California at Irvine
July 2004
© 2004 P.JorionE-mail: [email protected]
VAR
Please do not reproducewithout author’s permission
Measuring Market Risk
411-ecs60931.swf; VARstart.swf
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Measuring Market Risk:
PLAN(1) Risk factors and mapping
(2) Approaches to VAR
(3) Modeling time-variation in risk
(4) The Basel Internal Model Approach
Measuring Market Risk
(1)
Risk Factors and Mapping
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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
Risk Decomposition:
“The Theory of Particle Finance”! Define risk factors
» bonds: first factor is yield change (duration model)
» stocks: first factor is market (diagonal model)
» forward contract: factor is spot exchange rate andinterest rates
! Decompose all positions as exposures on risk
factors! Aggregate all exposures across the portfolio
! Assess possible movements in risk factors
! Reconstruct risk of total portfolioRisk Management - Philippe Jorion
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Mapping in Risk Measurement
Risk Management - Philippe Jorion 424-ecs41433.swf; VARmapping.swf
Mapping in Risk MeasurementInstruments
#2#1 #5 #6#3 #4
#1 #2 #3
Risk Aggregation
RiskFactors
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Measuring Market Risk
(2)
Approaches to VAR
Approaches to VAR
Risk Management - Philippe Jorion
Monte CarloSimulations
QuadraticModels
LinearModels
HistoricalSimulations
Local ValuationMethods
Full ValuationMethods
Risk Measurement
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Approaches to VAR:
Local versus Full Valuation! In general, the portfolio value is a non-linear
function of risk factors V=V(S)
! Local valuation:
» price the portfolio at current position andcompute local derivatives V 0 !
» linear approximation: dV = #0 dS » simple and fast
! Full valuation:» reprice portfolio: dV = V(S 1 )-V(S 0 )
» much more time intensiveRisk Management - Philippe Jorion
0
V
S
$# %
$
Local Valuation Method! Replaces all positions
by a portfolio of delta(linear) exposures onrisk factors
! Assumes risk factorshave normal distribution
! Portfolio risk obtainedfrom delta exposuresand covariance matrix
Risk Management - Philippe Jorion 424-ecs?.swf; VARlocal.swf
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Local Valuation Method! Replaces all positions
by a portfolio of delta(linear) exposures onrisk factors
! Assumes risk factorshave normal distribution
! Portfolio risk obtainedfrom delta exposuresand covariance matrix
Risk Management - Philippe Jorion
Risk Factor
Payoff
S 0
V(S 0 )
#0
Full Valuation Method! Reprices all positions
under new values forrisk factors
! Assumes a distributionfor risk factors
! Portfolio risk obtainedfrom distribution ofportfolio values
Risk Management - Philippe Jorion 424-ecs?.swf; VARfull.swf
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Full Valuation Method! Reprices all positions
under new values forrisk factors
! Assumes a distributionfor risk factors
! Portfolio risk obtained
from distribution ofportfolio values
Risk Management - Philippe Jorion
Payoff
Risk Factor S 0
V(S 0 )
V(S 1 )
S 1
Approaches to VAR
! Delta-Normal
» combines linear positions with covariances
! Historical Simulation
» replicates current portfolio over historical data
! Monte Carlo Simulation
» creates simulations of financial variables
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Delta-Normal Method:
Example of a Forward Contract
Risk Management - Philippe Jorion
DomesticCurrency Bond
ForeignCurrency Bond
RiskFactor #2:RiskFactor #1:
Spot Price
ForwardContract
RiskFactor #3:
$16,392,393 $16,392,393 -$16,298,812
Delta-Normal Method:
Example
Risk Management - Philippe Jorion SL 40:”DeltaN” sheet in bpforws.xls (paste as wks)
Confidence level (%)
= 95 95%
Delta-normal VARResult = $127,148
Distribution of P&L
0
5
10
15
20
25
-$200,000 -$100,000 $0 $100,000 $200,000
P & L
F r e q u e n c y
Normal
VAR-Normal
Note: Change any of the inputs by entering a value or moving the scroll bar. Graph will automatically update.
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Delta-Normal Method:
Pros and Cons! Advantages:
» simple method
» fast computation, even for large portfolios
» can be extended to time-varying risk
» easy to explain
! Problems:
» linear model: may mismeasure risk of options» relies on normal approximation: cannot explain
“fat” tails
Risk Management - Philippe Jorion
Approaches to VAR:
Historical-Simulation! Assumptions:
» recent historical data relevant
» full valuation
! Method:
» use history of changes in risk factors #yi» starting from current values, construct yt+#yi ...
» evaluate portfolio under simulated factor
» compile distribution of portfolio changes
» “bootstrapping” method
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Historical-Simulation Method:
Example
Risk Management - Philippe Jorion SL 45:”HistSim” sheet in bpforws.xls (paste as wks)
Day
= 1 1
10
Portfolio return
Result = -$33,640
Confidence level (%)
= 95 95%
VAR
Result = $119,905
Historical Simulation of P&L
-$200,000
-$150,000
-$100,000
-$50,000
$0
$50,000
$100,000
$150,000
$200,000
98/08/10 98/09/08 98/10/06 98/11/03 98/12/01 98/12/31
Day
P r o f i t a n d
L o s s
Note: Change any of the inputs by entering a value or moving the scroll bar. Graph will automatically update.
VAR
Historical-Simulation Method:
Example
Risk Management - Philippe Jorion SL 46:”HistDist2” sheet in bpforws.xls (paste as wks)
Confidence level (%)= 95 95%
Historical-simulation VAR
Result = $119,905
Delta-normal VARResult = $127,148
Distribution of P&L
0
5
10
15
20
25
-$200,000 -$100,000 $0 $100,000 $200,000
P & L
F r e q u e n c y
Historical
NormalVAR-HS
VAR-Normal
Note: Change any of the inputs by entering a value or moving the scroll bar. Graph will automatically update.
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Historical-Simulation Method:
Pros and Cons! Advantages:
» accounts for non-normal data
» full valuation method
» easy to explain
! Problems:
» only one sample path, which may not be
representative» no time-variation in risk
Risk Management - Philippe Jorion
Approaches to VAR:
Monte Carlo! Assumptions:
» define joint stochastic model for risk factors
» full valuation
! Method:
» use numerical simulations for risk factors tohorizon
» value portfolio
» report full portfolio distribution
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Monte Carlo Simulation Method:
Example
Risk Management - Philippe Jorion SL 48:”MCSim” sheet in bpforws.xls (paste as wks)
Simulated Risk Factor
1.65
1.66
1.67
1.68
0 1Time
Note: Change any of the inputs b y entering a value or moving the scroll b ar. Graph will automatically update.
Distribution of P&L
-$200,000-$100,000 $0 $100,000 $200,000
P&L
F r e q u e n c y
Monte Carlo Simulation Method:
Example
Risk Management - Philippe Jorion SL 49:”MCHistDist2” sheet in bpforws.xls (paste as wks)
Confidence level (%)
= 95 95%
Monte Carlo-simulation VAR
Result = $132,669
Delta-normal VAR
Result = $127,148
Distribution of P&L
-$200,000 -$100,000 $0 $100,000 $200,000
P&L
F r e q u e n c y
Monte Carlo
VAR
Note: Change any of the inputs b y entering a value or moving the scroll b ar. Graph will automatically update.
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Monte Carlo Method:
Pros and Cons! Advantage:
» most flexible method
» appropriate for complex instruments
» allows fat tails and time-variation in risk
! Problems:
» computational cost
» most difficult to implement--intellectual cost» subject to “model risk”--wrong assumptions
» subject to sampling estimation error
Risk Management - Philippe Jorion
Approaches to VAR:
Comparison
Risk Management - Philippe Jorion
Delta-normal Historical-simulation
Monte- Carlo
Valuation Linear Nonlinear Nonlinear
Distribution Normal,Time-varying
Actual General
Speed Fastest Fast Slow
Pitfalls Options,Fat tails
Short sample Model risk,Sampling error
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Approaches to VAR:
FSA Survey
Risk Management - Philippe Jorion
HistoricalSimulation,
31%
Delta
Normal,
42%
MC
Simulation,
23%
Measuring Market Risk
(3)
Modeling time variation in risk
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Time Variation in Risk! There is strong evidence that daily volatility
moves in a predictable fashion for mostfinancial series
! Risk measures can be adapted to model timevariation, based on historical data
! Time series models for volatility can also pick
up structural changes (e.g. transition fromfixed to flexible exchange rate system)
Risk Management - Philippe Jorion
Volatility Estimation:
(1) Moving Average! Define the innovation as squared daily return
x(t) = R2t
! Using a moving window of size N , thevariance forecast is:
!
The volatility forecast is & t = )ht ! Recent large movements will increase the
variance forecast, as long as within thewindow (but drop off after N )
Risk Management - Philippe Jorion
2
1
1 N t t ii
h R N
*%% +
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! The variance forecast is:
» conditional residual , t =(Rt / & t ) is normal
» recursive forecast: history summarized in h
! Uses exponentially decaying weights:
» weights on older observations decreaseEWMA = Exponentially Weighted Moving Average
Risk Management - Philippe Jorion
Volatility Estimation:
(2) Exponential Smoothing
2 2 2 2
1 2 3(1 )[ ...]t t t t h R R R- - - * * *% * . . .
2
1 1(1 )t t t h R h- - * *% * .
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Weights on Previous Days: Daily Model
100 75 50 25 0
0.06
0.05
0.04
0.03
0.02
0.01
0
Exponential Model,Decay=0.97
Exponential Model,Decay=0.94
Moving Average Model,Window=60
Days in the PastRisk Management - Philippe Jorion
! Benefits:
» easy to implement--one parameter only
» should lead to positive definite covariance matrix
» special case of GARCH process--performs well
! Estimation
» example: JP Morgan RiskMetrics
» choice of decay factor, - =0.94 for all daily series
» however, cannot be extended to longer horizons
Risk Management - Philippe Jorion
EWMA
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Volatility Estimation:
(3) GARCH Models! More general time-series model, with realistic
persistence in volatilityGARCH= Generalized Autoregressive Conditional
Heteroskedasticity
! Typical GARCH(1) model:
ht = a0 + a1 Rt-12 + b ht-1
» long-run forecast is h = a0 / (1-a1-b)
» persistence parameter is (a1+b)» model can be extended to long horizon forecasts
» describes well most financial time seriesRisk Management - Philippe Jorion
1.5
1
0.5
0
Daily volatility
Exponential Model
GARCH Model
1990 1991 1992 1993 1994
Volatility Forecasts: $/BP Exchange Rate
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Historical Simulation with
Time-Varying Volatility! Fit GARCH model to time series
! Construct scaled residuals , t =(Rt / & t )
! Apply historical simulation to scaled residualand multiply by latest volatility forecast
! Example:» current & t =1.5%
» at t-20, Ri =1.6%, & i =1%, , i =1.6
» forecast R*t = , i ×& t = 1.6×1.5% = 2.1%
» repeat for all observations in the HS window(See Hull and White, Journal of Risk, Fall 1998)
Risk Management - Philippe Jorion
Capital Required for a Position of $1 in DMHS: Historical Simulation using 500 most recent observations
BRW: Historical Simulation with exponential weights
HW: Historical Simulation with volatility changes
Source: Hull and White
0
0.1
0.2
0.3
0.4
0.5
D e c - 8 9
J u n - 9 0
D e c - 9 0
J u n - 9 1
D e c - 9 1
J u n - 9 2
D e c - 9 2
J u n - 9 3
D e c - 9 3
J u n - 9 4
D e c - 9 4
J u n - 9 5
D e c - 9 5
J u n - 9 6
D e c - 9 6
J u n - 9 7
HS
BRW
HW
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1.5
1
0.5
01990 1991 1992 1993 1994
Volatility forecast (%)
1-dayforecast
1-yearforecast
Short- and Long-Term GARCH Forecast
Risk Management - Philippe Jorion
GARCH Models:
Major Issues! Little evidence of predictability in risk over
longer horizons, e.g. beyond one month
! Using fast-moving GARCH system wouldcreate capital charges that fluctuate toomuch
! Basel Committee disallows GARCH models
(minimum window is one year)
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Measuring Market Risk
(4)
The Basel
Internal Model Approach
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 ofhistorical 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)'i=160 VAR(t-i), VAR(t-1)]+SRC
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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
Risk Management - Philippe Jorion
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, 1failure very 4 years)
! Plus factor: a penalty component shall beadded to k if back-testing reveals that the
bank's internal model incorrectly forecastsrisks, or internal risk management practicesare viewed as “inadequate”
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Why the Multiplicative Factor?! To protect against model risk, or “fat tails”
! For any random variable x with finitevariance, Chebyshev’s inequality states» P[|x-/|>r &] 0 1/r 2
» if symmetric, P[(x-/1
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VAR Reporting: 2003
Risk Management - Philippe Jorion \var\wbank\var-annual.xls (paste as PIC)
Institution Conf. Capital
LevelReported 99% General Actual ($MM)
US Commercial Banks
Bank of America 99 34 34 319 - 66,651
Citicorp 99 39 39 370 816 76,153
JP Morgan Chase 99 175 175 1,659 - 59,816
US Investment Banks
Goldman Sachs 95 58 82 778 21,362
Merrill Lynch 95 27 39 366 27,651
Morgan Stanley 99 58 58 550 24,867
Non-US Commercial Banks
Deutsche Bank 99 61 61 576 956 37,447
UBS 99 90 90 853 1,174 26,979
Barclays 98 46 52 495 1,973 43,110(Annual average)
1-day VAR ($MM) MRC ($MM)
Informativeness of VAR:Realized and Forecast Risk (8 US Banks, 95.Q1-00.Q3)
Risk Management - Philippe Jorion
$0
$100
$200
$300
$400
$500
$600
$700
$800
$900
$0 $50 $100 $150 $200 $250
Absolute va lue of un ex pected trad ing reve nu e
VAR-based risk forecast
Jorion (2002), ̀ `How Informative are Value-at-Risk Disclosures?,'' Accounting Review
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The “Puzzle” of
Conservativeness of VAR Measures
! Reported VARs are “too large”:» possibly because capital adequacy requirements
are not binding, or to avoid regulatory intrusion
P&L VAR Excess
99th Pc Mean of VAR Obs Exp Nb Mean
Bank 1 -1.78 -1.87 5% 569 6 3 -2.12
Bank 2 -2.26 -1.74 -23% 581 6 6 -0.74
Bank 3 -2.73 -4.41 62% 585 6 3 -3.18
Bank 4 -1.59 -5.22 228% 573 6 0 NA
Bank 5 -2.78 -5.62 102% 746 7 1 -0.78
Bank 6 -0.97 -1.72 77% 586 6 3 -5.80
Exceptions
Comparison of P&L Percentile a nd VAR
Source: Berkowitz and O'Brien (2002), “How Accurate are the Value-at-Risk Models at Commercial Banks,” Journal of Finance
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Measuring Market Risk
(5) Conclusions
CONCLUSIONS (1)! Market risk measurement applies to large-
scale portfolio and requires simplifications
! Among major design choices are
(1) the choice and number of risk factors
(2) the choice of a local versus full valuationmethod for the instruments
! These choices depend on the nature of theportfolio and reflect tradeoffs between speedand accuracy
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CONCLUSIONS (2)! The ultimate goal of risk measurement is to
understand risk better so as to manage iteffectively
! Risk management should not only preventlosses, but add value to the decision process
! Tools such as marginal and component VARare integral to portfolio management
! Proper risk management requires competentrisk managers
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