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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
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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 SEMIVARIANCE
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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
K e r n e l D e n s ity ( N o r m a l , h =
0 .1 1 4 5 )
0 .8
0 .6
0 .4
0 .2
0 .0
- 20
- 15
- 10
S & P 5 00
-5
%
R E T U R N S
0
5
CUMULATIVE
DISTRIBUTION
Empirical CDF of S&P500 RETURNS
1.0
0.8
0.6
0.4
0.2
0.0
-20
-10
0
10
Weakness of this measure
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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

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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 st  wt ' H t wt
And assuming normality, VaRt=-1.64 st
Quality of VaR depends upon H and
normality assumption.
PORTFOLIO MODELS

Historical performance of fixed
weight portfolio is calculated from
data bank
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Model for quantile is estimated

VaR is forecast
COMPLICATIONS

Some assets didn’t trade in the pastapproximate by deltas or betas

Some assets were traded at different
times of the day - asynchronous pricessynchronize these

Derivatives may require special
assumptions - volatility models and
greeks.
PORTFOLIO MODELS EXAMPLES


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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
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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
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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)
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Runs - Mood (1940)
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Ljung Box on hits (1979)
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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
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If out of sample test , or
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If all parameters are known
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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_HIT
Sample: 5 2892
Included observations: 2888
Variable
Coefficient Std. Error t-Statistic
Prob.
C
SAV_HIT(-1)
SAV_HIT(-2)
SAV_HIT(-3)
SAV_HIT(-4)
SAV_VAR
0.0051
0.0397
0.0244
0.0252
-0.0044
-0.0034
0.5977
0.0334
0.1920
0.1781
0.8127
0.6002
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.0029
0.0012
0.2190
138.2105
291.2040
1.9999
0.0096
0.0187
0.0187
0.0187
0.0187
0.0066
0.5277
2.1277
1.3051
1.3468
-0.2370
-0.5241
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.0006
0.2191
-0.1975
-0.1851
1.7043
0.1301
Some Extensions
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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
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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
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VaR is a function of observables in t-1
VaR=f(VaR(t-1), y(t-1), parameters)
For example - the Adaptive Model
VaR  VaR   (hit )
t 1
t
t 1
hit  I ( y  VaR )  
t
t
t
How to compute VaR
If beta is known, then VaR can be
calculated for the adaptive model from a
starting value.
Let VaR(1)  1.65
  * .95 if hit in 1
VaR(2)  VaR(1)  
  * (-.05) if no hit
VaR (3)  .....
CAViaR News Impact Curve
More Specifications

Proportional Symmetric Adaptive
VaRt  VaRt 1   1( yt 1  VaRt 1 )   2 ( yt 1  VaRt 1 )

Symmetric Absolute Value:
VaRt 1   0   1VaRt 1   2 yt 1
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Asymmetric Absolute Value:
VaRt 1   0   1VaRt 1   2 yt 1   3
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Asymmetric Slope

VaRt   0   1VaRt 1   2 yt 1   3 yt 1

Indirect GARCH


VaR


2
t

1
VaRt  k   0   1 
  2 yt 1 



k




2
1/ 2

REMAINING PROBLEMS
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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
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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
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$1Million Portfolio at a point in time23,2000
Find 1% VaR
Construct historical portfolio
March
– 50% Nasdaq, 30%DowJones,20% LongBonds

Build GARCH
– Compute VaR - Gaussian, Semiparametric

Estimate CAViaR
PORTFOLIO COMPONENTS
0.10
0.05
0.00
-0.05
-0.10
3/27/90
2/25/92
1/25/94
12/26/9 5
11/25/9 7
10/26/9 9
11/25/9 7
10/26/9 9
11/25/9 7
10/26/9 9
NQ
0.10
0.05
0.00
-0.05
-0.10
3/27/90
2/25/92
1/25/94
12/26/9 5
DJ
0.10
0.05
0.00
-0.05
-0.10
3/27/90
2/25/92
1/25/94
12/26/9 5
RA T E
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-
CORRELATIONS
NQ
DJ
RATE
NQ
DJ
RATE
1.000000 0.695927 0.145502
0.695927 1.000000 0.236221
0.145502 0.236221 1.000000
HISTORICAL QUANTILE

DECADE OF HISTORICAL DATA:
– VaR=$22600
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ONE YEAR OF HISTORICAL DATA:
– VaR=$24800

WORST LOSS OVER YEAR: $36300
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
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
0.025
0.020
0.015
0.010
0.005
0.000
3/26/90
1/24/94
11/24/97
CALCULATE VaR
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ASSUMING NORMALITY
– VaR=2.326348* 0.014605*1000000
– $33,977
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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
0.06
0.05
0.04
0.03
0.02
0.01
3/23/90
2/21/92
1/21/94
12/22/95
11/21/97
VAR _C AVIAR _OPT
10/22/99
CAViaR ESTIMATE
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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.