Value at Risk: Market Risk Models
Han Zhang
Director, Head of Market Risk Analytics
Corporate Market and Institutional Risk
August 23, 2013
University of North Carolina at Charlotte
Value at Risk
 What is VaR
In its most general form, the Value at Risk measures the potential
loss in value of a risky asset or portfolio over a defined period for a
given confidence interval.
 What does it mean
1-day 99% VaR at $10 mm for an Equity Portfolio
 How to calculate it
If the equity portfolio only has one stock from IBM, we collect for
example one year of stock price history on IBM, and calculate the
daily returns, and then apply such returns to the holding to
calculate the profit and loss (P&Ls), after that the vector of the
P&Ls are sorted in order, the 1% tail on loss side is the VaR.
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List of Models in Risk Analytics (RA)
 General Value at Risk (General VaR)
 Stress General VaR
 Debt Specific Risk (DSR)
 Equity Specific Risk (ESR)
 Incremental Risk Charge (IRC)
 Stress Testing
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What are all these models doing
 General VaR captures the risk from general market moves;
 Stress General VaR follows the same methodology as the
General VaR, but it captures the risk from a most stressful time
period;
 Additional to General VaR, DSR and ESR model capture the
idiosyncratic moves from individual names, plus the event risk
specific to the name;
 IRC captures default and migration risk beyond 10-day and to
1-year specific to the name;
 Bank conducts stress tests (forward looking assessments) to
gain information about the impact of adverse market events on
its positions.
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Model Coverage and Usage (Trading)
 Covers all covered trading positions:
– Interest rate
– Equity
– Commodity
– FX
– Credit products
– Structure products
 The models are used by
– Market Risk Oversight to monitor the risks/limits for trading
desk
– Market Risk Reporting Team to report bank’s Regulatory
Capital
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VaR method
 VaR method
– Variance - Covariance
– Historical Simulation
– Monte Carlo Simulation
 P&L calculation method
– Delta-gamma approximation
– Grid approximation
– Full revaluation
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How we monitor the VaR model - Backtesting
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Debt Specific Risk Model
 It captures the idiosyncratic risk to specific names.
 The idiosyncratic risk can be captured by two ways:
– Conditional on no market component move;
– Use regression to decompose the total risk into
market/systematic component and idiosyncratic component.
rj   j rI Kj   j  j
2
 total
  j2  I2, KK   2j
j
 Monte Carlo Simulation is used for DSR
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Equity Specific Risk Model
 Bank’s general VaR captures idiosyncratic risk and some
of the event risk if it happens in last one year window;
 ESR captures the event risk and default risk
 We use a Stochastic Variance VaR model to capture the
heavy tail
X (t )  AZ I (t ), Z iI (t )  Vi (t )Z i , Z i ~ N (0, 1), i  1,2,, M .
L
L
l 1
l 1
Vi (t )  Bi  (t )   bil l (t ),  bil  1, bil  0, i  1,2,, M , l  1,2,, L
 l (T ) ~ ( l ,  l , l )
 The default model is captured by the credit event
simulation model
 Monte Carlo simulation is used for ESR also
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Credit Event Simulation Model in IRC
 Wells Fargo’s simulation model for credit events
(migrations and defaults) is similar to the
CreditMetrics methodology
 Historical transition matrix driven the migration and
default probabilities
AAA
AA
A
BBB
BB
B
CCC
D
AAA
0.9123
0.0058
0.0004
0.0001
0.0002
0
0
0
AA
A
BBB
BB
B
CCC
D
0.0799
0.0054
0.0006
0.0008
0.0003
0.0006
0
0.9021
0.0844
0.0057
0.0006
0.0009
0.0002
0.0002
0.0205
0.9147
0.0575
0.0042
0.0017
0.0002
0.0008
0.0015
0.0403
0.9018
0.0443
0.0075
0.0017
0.0028
0.0006
0.002
0.0573
0.8374
0.0829
0.0088
0.0108
0.0005
0.0017
0.0027
0.0616
0.8249
0.0527
0.0559
0
0.0025
0.0036
0.0103
0.1317
0.5252
0.3267
0
0
0
0
0
0
1
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Credit Event Simulation Model in IRC
 Under the assumption of independence of credit
events for all issuers, the transition matrix gives
proper migration and default probabilities separately
for each name.
10
,,
,
Credit Event Simulation Model in IRC
 Modeling of correlated asset returns
– Model should take into account the migration and default
correlations among all issuers
Z  AS w
CorrS  AS AS'
– A CAPM-type model is used to decrease the dimension of the
correlation matrix
rj   j rI Kj   j  j
rI  BI x
C I  BI BI'
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