RISK MANAGEMENT

GOALS AND TOOLS

ROLE OF RISK MANAGER

MONITOR RISK OF A FIRM, OR OTHER ENTITY– IDENTIFY RISKS– MEASURE RISKS– REPORT RISKS– MANAGE -or CONTROL RISKS

COMMON TYPES OF RISK

MARKET RISK CREDIT RISK LIQUIDITY RISK OPERATIONAL RISK SYSTEMIC RISK

COMMON TOOLS

SCENARIO ANALYSIS– ASSESS IMPLICATIONS OF PARTICULAR

COMBINATIONS OF EVENTS– NO PROBABILITY STATEMENT

STATISTICAL ANALYSIS– FIND PROBABILITY OF LOSSES– HOW TO ASSESS EVENTS WHICH HAVE

NEVER OCCURRED?

STATISTICAL ANALYSIS OF MARKET RISK PORTFOLIO STANDARD DEVIATION

DOWNSIDE RISK SUCH AS SEMI-VARIANCE

VALUE AT RISK

Value at Risk is a single measure of market risk of a firm, portfolio, trading desk, or other economic entity.

It is defined by a confidence level and a horizon. For convenience consider 95% and 1 day.

A ny loss tomorrow will be less than the Value at Risk with 95% certainty

HISTOGRAM OF TOMORROW’S VALUE - BASED ON PAST RETURNS

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- 2 0 - 1 5 - 1 0 - 5 0 5

S & P 5 0 0 % R E T U R N S

K e r n e l D e n s i t y ( N o r m a l , h = 0 . 1 1 4 5 )

CUMULATIVE DISTRIBUTION

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Empirical CDF of S&P500 RETURNS

Weakness of this measure

The amount we exceed VaR is important There is no utility function associated with

this measure The measure assumes assets can be

sold at their market price - no consideration for liquidity

But it is simple to understand and very widely used.

THE PROBLEM

FORECAST QUANTILE OF FUTURE RETURNS

MUST ACCOMMODATE TIME VARYING DISTRIBUTIONS

MUST HAVE METHOD FOR EVALUATION

MUST HAVE METHOD FOR PICKING UNKNOWN PARAMETERS

TWO GENERAL APPROACHES FACTOR MODELS--- AS IN

RISKMETRICS

PORTFOLIO MODELS--- AS IN ROLLING HISTORICAL QUANTILES

FACTOR MODELS

– Volatilities and correlations between factors are estimated

– These volatilities and correlations are updated daily

– Portfolio standard deviations are calculated from portfolio weights and covariance matrix

– Value at Risk computed assuming normality

FACTOR MODEL: EXAMPLE

If each asset is a factor, then an nxn covariance matrix, Ht ,is needed.

LET wt be the portfolio weights on day t Then standard deviation is And assuming normality, VaRt=-1.64 st

Quality of VaR depends upon H and normality assumption.

tttt wHws '

PORTFOLIO MODELS

Historical performance of fixed weight portfolio is calculated from data bank

Model for quantile is estimated

VaR is forecast

COMPLICATIONS

Some assets didn’t trade in the past- approximate by deltas or betas

Some assets were traded at different times

of the day - asynchronous prices-

synchronize these

Derivatives may require special

assumptions - volatility models and greeks.

PORTFOLIO MODELS - EXAMPLES Rolling Historical : e.g. find the 5%

point of the last 250 days GARCH : e.g. build a GARCH model to

forecast volatility and use standardized residuals to find 5% point

Hybrid model: use rolling historical but weight most recent data more heavily with exponentially declining weights.

GARCH EXAMPLE

Choose a GARCH model for portfolio Forecast volatility one day in advance Calculate Value at Risk

– Assuming Normality, multiply standard deviation by 1.64 for 5% VaR

– Otherwise (and better) calculate 5% quantile of standardized residuals as factor

Multi-day forecasts: what distribution to use?

DIAGNOSTIC CHECKS

Define hit= I(return<-VaR)-.05 Percentage of positive hits should not be

significantly different from theoretical value

Timing should be unpredictable VaR itself should have no value in

predicting hits TESTS?

Tests

Cowles and Jones (1937)

Runs - Mood (1940)

Ljung Box on hits (1979)

Dynamic Quantile Test

Dynamic Quantile Test

To test that hits have the same distribution regardless of past observables

Regress hit on– constant– lagged hits– Value at Risk– lagged returns– other variables such as year dummies

Distribution Theory

If out of sample test , or If all parameters are known

Then TR02 will be asymptotically Chi

Squared and F version is also available But the distribution is slightly different

otherwise

Dynamic Quantile Test -SP

Dependent Variable: SAV_HITSample: 5 2892Included observations: 2888Variable Coefficient Std. Error t-Statistic Prob.

C 0.0051 0.0096 0.5277 0.5977SAV_HIT(-1) 0.0397 0.0187 2.1277 0.0334SAV_HIT(-2) 0.0244 0.0187 1.3051 0.1920SAV_HIT(-3) 0.0252 0.0187 1.3468 0.1781SAV_HIT(-4) -0.0044 0.0187 -0.2370 0.8127SAV_VAR -0.0034 0.0066 -0.5241 0.6002

R-squared 0.0029 Mean dependent var 0.0006Adjusted R-squared 0.0012 S.D. dependent var 0.2191S.E. of regression 0.2190 Akaike info criterion -0.1975Sum squared resid 138.2105 Schwarz criterion -0.1851Log likelihood 291.2040 F-statistic 1.7043Durbin-Watson stat 1.9999 Prob(F-statistic) 0.1301

Some Extensions

Are there economic variables which can predict tail shapes?

Would option market variables have predictability for the tails?

Would variables such as credit spreads prove predictive?

Can we estimate the expected value of the tail?

THE CAViaR STRATEGY

Define a quantile model with some unknown parameters

Construct the quantile criterion function Optimize this criterion over the historical

period Formulate diagnostic checks for model

adequacy Read Engle and Manganelli

SPECIFICATIONS FOR VaR

VaR is a function of observables in t-1 VaR=f(VaR(t-1), y(t-1), parameters) For example - the Adaptive Model

)(

)(11

ttt

ttt

VaRyIhit

hitVaRVaR

How to compute VaR

If beta is known, then VaR can be calculated for the adaptive model from a starting value.

.....)3(

hit no if (-.05)*

1in hit if .95*VaR(1)VaR(2)

1.65VaR(1)Let

VaR

CAViaR News Impact Curve

More Specifications

Proportional Symmetric Adaptive

Symmetric Absolute Value:

Asymmetric Absolute Value:

)VaRy()VaRy(VaRVaR 1t1t21t1t11tt

1t21t101t yVaRVaR

31t21t101t yVaRVaR

Asymmetric Slope

Indirect GARCH

1t31t21t10t yyVaRVaR

2/12

1t2

21t

10t yk

VaRkVaR

REMAINING PROBLEMS

Other Risks, I.e. credit and liquidity risk Derivatives are not easy in either approach

– Approximate by delta and ignore volatility risk?– Simulate and reprice using BS?– Use simulation of simulations– Longstaff&Schwarz clever idea

• one simulation plus a regression.

RISK MANAGEMENT

IN MEAN VARIANCE WORLD, RISK MANAGEMENT DOES NOT EXIST AS A SEPARATE PROBLEM, MERELY COORDINATION.

COULD MAXIMIZE UTILITY s.t. VaR CONSTRAINT.

RISK REDUCTION CAN BE A MEAN VARIANCE PROBLEM ITSELF.

Value at Risk: A Case Study

$1Million Portfolio at a point in time- March 23,2000

Find 1% VaR Construct historical portfolio

– 50% Nasdaq, 30%DowJones,20% LongBonds Build GARCH

– Compute VaR - Gaussian, Semiparametric Estimate CAViaR

PORTFOLIO COMPONENTS

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NQ

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DJ

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RATE

STATISTICS

NQ DJ RATE

Mean 0.000928 0.000542 0.000137

Median 0.001167 0.000281 0.000000

Maximum 0.058479 0.048605 0.028884

Minimum -0.089536-0.074549-0.042677

Std. Dev. 0.011484 0.009001 0.007302

Skewness -0.530669-0.359182-0.202732

Kurtosis 7.490848 8.325619 4.956270

CORRELATIONS

NQ DJ RATE

NQ 1.000000 0.695927 0.145502

DJ 0.695927 1.000000 0.236221

RATE 0.145502 0.236221 1.000000

HISTORICAL QUANTILE

DECADE OF HISTORICAL DATA:– VaR=$22600

ONE YEAR OF HISTORICAL DATA:– VaR=$24800

WORST LOSS OVER YEAR: $36300

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PORT

Value at Risk by GARCH(1,1)

C 1.40E-06 4.48E-07 3.121004

ARCH(1) 0.077209 0.017936 4.304603

GARCH(1) 0.904608 0.019603 46.14744

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CALCULATE VaR

ASSUMING NORMALITY– VaR=2.326348* 0.014605*1000000– $33,977

ASSUMING I.I.D. DISTURBANCES– VaR=2.8437*0.014605*1000000– $ 39,996

CAViaR MODEL

MAXIMIZE QUANTILE CRITERION BY GRID SEARCH:

var=c(1)+c(2)*var(-1)+c(3)*abs(y)

c(1) =0.002441

c(2) =0.796289

c(3) =0.346875

VaR over TIME

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VAR_CAVIAR_OPT

CAViaR ESTIMATE

1% VaR is $38,228

This is very plausible - it is worse than the rolling quantiles as volatility was rising

It lies just below the semi-parametric GARCH.