assessing market risk by jorion philippe

8/18/2019 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


(1)

Components of riskmeasurement systems


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Philippe Jorion 3

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


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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Philippe Jorion 4

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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Philippe Jorion 5

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


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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Philippe Jorion 6

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


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


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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Philippe Jorion 9


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.




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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Philippe Jorion 10



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.



(2)

Value at Risk as a measure ofdownside risk


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Philippe Jorion 11

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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Philippe Jorion 12

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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Philippe Jorion 13

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


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/$)


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


110 /]($)[($) ###% t t t t S S S Q R


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Philippe Jorion 15

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


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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Philippe Jorion 16

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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Philippe Jorion 17


2003 Annual Report (5)Value-at-Risk: average $69 million


$69m





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Philippe Jorion 18


(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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Philippe Jorion 20

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



(4)

VAR caveats:

Alternative measures of risk


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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


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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Philippe Jorion 24


VAR does not describe the losses in the left tail

411-ecs13540.swf; VARsame.swf


» 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


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


VAR

VAR does not describe the losses in the left tail

Change to VARsame.swf


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Philippe Jorion 25


VAR is measured with some error

411-ecs18141.swf; VARerror.swf


» 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


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


VAR

Change to VARerror.swf


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Philippe Jorion 26

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


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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Philippe Jorion 27


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


VAR

ETL

)1

(][

1

+%

%#, N

ii x

N VAR X ETL

Change to VARETL.swf


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Philippe Jorion 28

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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Philippe Jorion 29

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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Philippe Jorion 30

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


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


+%

##

%T

i

i x E xT

X

1

2)]([)1(

1)()


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Philippe Jorion 31


(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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Philippe Jorion 32

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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Philippe Jorion 33

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


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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Philippe Jorion 34

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


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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Philippe Jorion 35


2003 Annual Report (7)



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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Philippe Jorion 36


(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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Philippe Jorion 37

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


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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assessing market risk by jorion philippe

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