measuring market risk - webline.sfi.org.tw

Measuring Market Risk

Philippe Jorion 1


Philippe JorionUniversity of California at Irvine

July 2004

© 2004 P.JorionE-mail: [email protected]

VAR

Please do not reproduce without author�s permission


411-ecs60931.swf; VARstart.swf


Philippe Jorion 2

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


(1)Risk Factors and Mapping


Philippe Jorion 3

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

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 and

interest 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 portfolio



Philippe Jorion 4

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

Risk Factors



Philippe Jorion 5


(2)Approaches to VAR

Approaches to VAR


Monte Carlo Simulations

QuadraticModels

Linear Models

HistoricalSimulations

Local ValuationMethods

Full ValuationMethods

Risk Measurement


Philippe Jorion 6

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 and

compute local derivatives V0, » linear approximation: dV = ∆0 dS» simple and fast

! Full valuation:» reprice portfolio: dV = V(S1)-V(S0)» much more time intensive


0VS

∂∆ =

∂

Local Valuation Method

! Replaces all positions by a portfolio of delta (linear) exposures on risk factors

! Assumes risk factors have normal distribution

! Portfolio risk obtained from delta exposures and covariance matrix

Risk Management - Philippe Jorion 424-ecs?.swf; VARlocal.swf


Philippe Jorion 7

Local Valuation Method

! Replaces all positions by a portfolio of delta (linear) exposures on risk factors

! Assumes risk factors have normal distribution

! Portfolio risk obtained from delta exposures and covariance matrix


Risk Factor

Payoff

S0

V(S0)∆0

Full Valuation Method

! Reprices all positions under new values for risk factors

! Assumes a distribution for risk factors

! Portfolio risk obtained from distribution of portfolio values

Risk Management - Philippe Jorion 424-ecs?.swf; VARfull.swf


Philippe Jorion 8

Full Valuation Method

! Reprices all positions under new values for risk factors

! Assumes a distribution for risk factors

! Portfolio risk obtained from distribution of portfolio values


Payoff

Risk FactorS0

V(S0)

V(S1)

S1

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



Philippe Jorion 9

Example: We hold a position in a forward contract to buy 10MM British pounds (BP) for $16.5MM in three months. Based on the last 100 days, what is its 95%, 1-day VAR?


Plot:

Day= 100 100

ValueSpot rate ($/£) = $1.6637

Dollar rate (% pa) = 4.9375Pound rate (% pa) = 5.9688

Historical Data

1.56

1.58

1.60

1.62

1.64

1.66

1.68

1.70

1.72

1.74

98/08/10 98/09/08 98/10/06 98/11/03 98/12/01 98/12/31Day

Varia

ble

Note: Change any of the inputs by entering a value or moving the scroll bar. Graph will automatically update.

Spot rateDollar interest ratePound interest rate

SL 40:�DATA� sheet in bpforws.xls (paste as wks)ActionSetting:On Click, open object

Approaches to VAR:Delta-Normal

! Assumptions:» returns are normally distributed» payoffs are linear in the risk factors

! Method:» portfolio variance combines linear positions

with forecast of covariance matrixσp2 = w�t Σt+1 wt

VAR = α σp W



Philippe Jorion 10

Delta-Normal Method:Example of a Forward Contract


Domestic Currency Bond

Foreign Currency 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

Freq

uenc

y

Normal

VAR-Normal



Philippe Jorion 11

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


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



Philippe Jorion 12

Historical-Simulation Method:Example

Risk Management - Philippe Jorion SL 45:�HistSim� sheet in bpforws.xls (paste as wks)

Day= 1 1

10

Portfolio returnResult = -$33,640


VARResult = $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 98/12/31Day

Prof

it an

d Lo

ss


VAR

Historical-Simulation Method:Example

Risk Management - Philippe Jorion SL 46:�HistDist2� sheet in bpforws.xls (paste as wks)


Historical-simulation VARResult = $119,905


Distribution of P&L

0

5

10

15

20

25

-$200,000 -$100,000 $0 $100,000 $200,000

P & L

Freq

uenc

y

HistoricalNormalVAR-HS

VAR-Normal



Philippe Jorion 13

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


Approaches to VAR:Monte Carlo

! Assumptions:» define joint stochastic model for risk factors» full valuation

! Method:» use numerical simulations for risk factors to

horizon» value portfolio » report full portfolio distribution



Philippe Jorion 14

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


Distribution of P&L

-$200,000-$100,000 $0 $100,000 $200,000P&L

Freq

uenc

y

Monte Carlo Simulation Method:Example

Risk Management - Philippe Jorion SL 49:�MCHistDist2� sheet in bpforws.xls (paste as wks)


Monte Carlo-simulation VARResult = $132,669


Distribution of P&L

-$200,000 -$100,000 $0 $100,000 $200,000

P&L

Freq

uenc

y

Monte Carlo

VAR



Philippe Jorion 15

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


Approaches to VAR:Comparison


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


Philippe Jorion 16

Approaches to VAR:FSA Survey


Historical Simulation,

31%

Delta Normal,

42%

MC Simulation,

23%


(3)Modeling time variation in risk


Philippe Jorion 17

Time Variation in Risk! There is strong evidence that daily volatility

moves in a predictable fashion for most financial series

! Risk measures can be adapted to model time variation, based on historical data

! Time series models for volatility can also pick up structural changes (e.g. transition from fixed to flexible exchange rate system)


Volatility Estimation: (1) Moving Average

! Define the innovation as squared daily returnx(t) = R2t

! Using a moving window of size N, the variance forecast is:

! The volatility forecast is σt = √ht

! Recent large movements will increase the variance forecast, as long as within the window (but drop off after N)


21

1 Nt t ii

h RN −=

= ∑


Philippe Jorion 18

! The variance forecast is:

» conditional residual εt=(Rt /σt) is normal» recursive forecast: history summarized in h

! Uses exponentially decaying weights:

» weights on older observations decreaseEWMA = Exponentially Weighted Moving Average


Volatility Estimation: (2) Exponential Smoothing

2 2 2 21 2 3(1 )[ ...]t t t th R R Rλ λ λ− − −= − + + +

21 1(1 )t t th R hλ λ− −= − +



Philippe Jorion 19

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, λ =0.94 for all daily series» however, cannot be extended to longer horizons


EWMA


Philippe Jorion 20

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 series


1.5

1

0.5

0

Daily volatility

Exponential Model

GARCH Model

1990 1991 1992 1993 1994

Volatility Forecasts: $/BP Exchange Rate



Philippe Jorion 21

Historical Simulation with Time-Varying Volatility

! Fit GARCH model to time series! Construct scaled residuals εt =(Rt /σt) ! Apply historical simulation to scaled residual

and multiply by latest volatility forecast! Example:

» current σt=1.5%» at t-20, Ri =1.6%, σi =1%, εi =1.6» forecast R*t = εi ×σt = 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 observationsBRW: Historical Simulation with exponential weightsHW: Historical Simulation with volatility changes

Source: Hull and White

0

0.1

0.2

0.3

0.4

0.5

Dec

-89

Jun-

90

Dec

-90

Jun-

91

Dec

-91

Jun-

92

Dec

-92

Jun-

93

Dec

-93

Jun-

94

Dec

-94

Jun-

95

Dec

-95

Jun-

96

Dec

-96

Jun-

97

HSBRWHW


Philippe Jorion 22

GARCH Models: Technical Issues

! Parameters need to be to estimated; risk of overfitting the model in sample

! Too complex for multivariate systems» number of parameters increases geometrically

with number of series, for all variances and pairwise covariances

» covariance matrix needs to be positive definite» this is why RiskMetrics uses same parameter λ

for all series! Mean reversion is typically fast--not useful

for horizons above 1 monthRisk Management - Philippe Jorion

0 1 5 10 15 20 25

1

0.5

0

Days ahead

0.986

0.950.90

0.80

Persistence parameter:

Initial shock

1.00

Variance

Average Variance

Persistence in GARCH Model



Philippe Jorion 23

1.5

1

0.5

01990 1991 1992 1993 1994

Volatility forecast (%)

1-day forecast

1-year forecast

Short- and Long-Term GARCH Forecast


GARCH Models: Major Issues

! Little evidence of predictability in risk over longer horizons, e.g. beyond one month

! Using fast-moving GARCH system would create capital charges that fluctuate too much

! Basel Committee disallows GARCH models (minimum window is one year)



Philippe Jorion 24


(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 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)Σi=1

60 VAR(t-i), VAR(t-1)]+SRC



Philippe Jorion 25

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


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�



Philippe Jorion 26

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/r2

» if symmetric, P[(x-µ)<-rσ] ≤ (1/2)1/r2

» set RHS to 1%; then r=7.071, VARMAX=7.071σ» with a normal distribution VARN=2.326σ» correction k= VARMAX/VARN=3.03

! There is arbitrariness in the joint choice of (c, T and k)


Internal Model:Equivalent Risk Charges

Horizon: 1 2 5 10 20 60 250Confidence

99.99% 5.9 4.2 2.7 1.9 1.3 0.77 0.38 Aaa99.9% 7.1 5.1 3.2 2.3 1.6 0.92 0.45 A399% 9.5 6.7 4.2 3.0 2.1 1.22 0.60 Baa397.5% 11.3 8.0 5.0 3.6 2.5 1.45 0.71 Ba395% 13.4 9.5 6.0 4.2 3.0 1.73 0.85 B190% 17.2 12.2 7.7 5.4 3.9 2.22 1.09 B2

84.1%(1xσ) 22.1 15.6 9.9 7.0 4.9 2.85 1.40 B3Normal and independent distribution



Philippe Jorion 27

VAR Reporting: 2003

Risk Management - Philippe Jorion \var\wbank\var-annual.xls (paste as PIC)

Institution Conf. CapitalLevelReported 99% General Actual ($MM)

US Commercial BanksBank of America 99 34 34 319 - 66,651Citicorp 99 39 39 370 816 76,153JP Morgan Chase 99 175 175 1,659 - 59,816

US Investment BanksGoldman Sachs 95 58 82 778 21,362Merrill Lynch 95 27 39 366 27,651Morgan Stanley 99 58 58 550 24,867

Non-US Commercial BanksDeutsche Bank 99 61 61 576 956 37,447UBS 99 90 90 853 1,174 26,979Barclays 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)


$0

$100

$200

$300

$400

$500

$600

$700

$800

$900

$0 $50 $100 $150 $200 $250

Absolute value of unexpected trading revenue

VAR-based risk forecastJorion (2002), ``How Informative are Value-at-Risk Disclosures?,'' Accounting Review


Philippe Jorion 28

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 Excess99th Pc Mean of VAR Obs Exp Nb Mean

Bank 1 -1.78 -1.87 5% 569 6 3 -2.12Bank 2 -2.26 -1.74 -23% 581 6 6 -0.74Bank 3 -2.73 -4.41 62% 585 6 3 -3.18Bank 4 -1.59 -5.22 228% 573 6 0 NABank 5 -2.78 -5.62 102% 746 7 1 -0.78Bank 6 -0.97 -1.72 77% 586 6 3 -5.80

ExceptionsComparison of P&L Percentile and VAR

Source: Berkowitz and O'Brien (2002), �How Accurate are the Value-at-Risk Models at Commercial Banks,� Journal of Finance


Philippe Jorion 29


(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 valuation

method for the instruments! These choices depend on the nature of the

portfolio and reflect tradeoffs between speed and accuracy



Philippe Jorion 30

CONCLUSIONS (2)! The ultimate goal of risk measurement is to

understand risk better so as to manage it effectively

! Risk management should not only prevent losses, but add value to the decision process

! Tools such as marginal and component VAR are integral to portfolio management

! Proper risk management requires competent risk managers


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) 824-5245FAX: (949) 824-8469

E-Mail: [email protected]: www.gsm.uci.edu/~jorion

measuring market risk - webline.sfi.org.tw

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