measuring market risk philippe jorion

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Measuring Market Risk

Philippe Jorion 1


Philippe JorionUniversity of California at Irvine

July 2004

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

VAR

Please do not reproducewithout author’s permission


411-ecs60931.swf; VARstart.swf


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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


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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 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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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

RiskFactors



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Philippe Jorion 5


(2)

Approaches to VAR

Approaches to VAR


Monte CarloSimulations

QuadraticModels

LinearModels

HistoricalSimulations

Local ValuationMethods

Full ValuationMethods

Risk Measurement


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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 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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Philippe Jorion 7

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 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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Philippe Jorion 8

Full Valuation Method! Reprices all positions

under new values forrisk factors

! Assumes a distributionfor risk factors

! Portfolio risk obtained

from distribution ofportfolio values


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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Philippe Jorion 10

Delta-Normal Method:

Example of a Forward Contract


DomesticCurrency Bond

ForeignCurrency Bond

RiskFactor #2:RiskFactor #1:

Spot Price

ForwardContract

RiskFactor #3:

$16,392,393 $16,392,393 -$16,298,812


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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Philippe Jorion 11


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



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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 return

Result = -$33,640


= 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


VAR


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



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Philippe Jorion 13


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 tohorizon

» value portfolio

» report full portfolio distribution



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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

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)


= 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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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


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Philippe Jorion 16

Approaches to VAR:

FSA Survey


HistoricalSimulation,

31%

Delta

Normal,

42%

MC

Simulation,

23%


(3)

Modeling time variation in risk


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Philippe Jorion 17

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)


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 )


2

1

1 N t t ii

h R N

*%% +


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Philippe Jorion 18

! 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



(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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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, - =0.94 for all daily series

» however, cannot be extended to longer horizons


EWMA


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Philippe Jorion 20


(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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Philippe Jorion 21

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)


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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Philippe Jorion 23

1.5

1

0.5

01990 1991 1992 1993 1994

Volatility forecast (%)

1-dayforecast

1-yearforecast

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 wouldcreate capital charges that fluctuate toomuch

! Basel Committee disallows GARCH models

(minimum window is one year)



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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 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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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, 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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Philippe Jorion 26

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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Philippe Jorion 27

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)


$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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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 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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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 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


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

measuring market risk philippe jorion

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