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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
0 . 0
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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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1.0
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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
-0.10
-0.05
0.00
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3/27/90 2/25/92 1/25/94 12/26/95 11/25/97 10/26/99
NQ
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-0.05
0.00
0.05
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3/27/90 2/25/92 1/25/94 12/26/95 11/25/97 10/26/99
DJ
-0.10
-0.05
0.00
0.05
0.10
3/27/90 2/25/92 1/25/94 12/26/95 11/25/97 10/26/99
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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3/23/90 2/21/92 1/21/94 12/22/95 11/21/97 10/22/99
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
0.000
0.005
0.010
0.015
0.020
0.025
3/26/90 1/24/94 11/24/97
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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3/23/90 2/21/92 1/21/94 12/22/95 11/21/97 10/22/99
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.