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.

1

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 2

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.

3

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 4

VaR method

  VaR method – Variance - Covariance – Historical Simulation – Monte Carlo Simulation P&L calculation method – Delta-gamma approximation – Grid approximation – Full revaluation 5

How we monitor the VaR model - Backtesting

6

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.

r j

 

j r I j K

 

j



j

 2

total j

 

j

2  2

I

,

KK

 

j

2  Monte Carlo Simulation is used for DSR 7

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 i I

(

t

) 

V i

(

t

)

Z i

,

Z i

~

N

( 0 , 1 ),

i

 1 , 2 ,  ,

M

.

 

V i

(

t

) 

B i

 (

t

) 

l L

  1

b il



l

(

t

) ,

l L

  1

b il

 1 ,

b il

 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 8

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 0.0799

0.9021

0.0205

0.0015

0.0006

0.0005

0 0 A 0.0054

0.0844

0.9147

0.0403

0.002

0.0017

0.0025

0 BBB 0.0006

0.0057

0.0575

0.9018

0.0573

0.0027

0.0036

0 BB 0.0008

0.0006

0.0042

0.0443

0.8374

0.0616

0.0103

0 B 0.0003

0.0009

0.0017

0.0075

0.0829

0.8249

0.1317

0 CCC 0.0006

0.0002

0.0002

0.0017

0.0088

0.0527

0.5252

0 D 0 0.0002

0.0008

0.0028

0.0108

0.0559

0.3267

1 9

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



A S w Corr S



A S A S

' – A CAPM-type model is used to decrease the dimension of the correlation matrix

r j

 

j r I j K r I



B I x

 

j



j C I



B I B I

' 11