Presented in joint seminar with MathWorks at July 5, 2012 (logo-free version)

1. Introduction of VAR/GVAR Model as a
Methodology to Develop Stress Test
Scenarios for Market Risks
Motoharu Dei
Milliman, Inc.
July 5, 2012
VAR = Vector Autoregression, GVAR = Global Vector Autoregression

2. Table of Contents
 Introduction
 What is VAR model
 Flow to implement stress tests using VAR model
 Benefits to use VAR model
 Challenges to model VAR
 Experience of VAR model
 GVAR model
 Image of implementation
 Appendix
VAR = Vector Autoregression, GVAR = Global Vector Autoregression

3. Introduction
 “Stress test”
– Insurance inspection manual of FSA of Japan fully revised in February 2011 describes use of
“stress test” as an item for review and evaluation of “asset management risk management
structure”.
– Stress test is sought to be used as a function to reinforce EC, which is focused by FSA in
constructing ERM.
 At the same time, specific methodologies for stress tests are unknown
– Thoughts presented in the inspection manual description
• “Market movement in large turmoil in the past”
• “Assume the worst situation”
• “Reflect risk characteristics of the relevant insurer”
• “When assumptions in the methodology for market risk measure are collapsed”
– Other publication showing FSA’s thought (“Release of partial revision of insurance inspection
manual (draft)”)
• “To review and evaluate the points if a company implements appropriate stress tests at the time
considering its size and characteristic and uses the results for specific judgment regarding risk
management”

4. I will introduce VAR/GVAR
as one of the technical solutions in introducing stress tests.
VAR = Vector Autoregression, GVAR = Global Vector Autoregression

5. What is VAR model
 VAR is originally a methodology commonly used to model macro economic indices in
the area of econometrics.
 VAR model means “vector autoregressive model”, where time-series variables of
autoregressive models (AR model) are made vector.
・・・
: Time-series variable vector : Constant term vector
: Coefficient matrix : (Normal) Noise vector
To set it as a macro economic index (e.g. domestic and foreign equity indices,
long- and short-term interest rates, price index)
 Projection model assuming that economic indices change while correlating each other
 Model naturally structured considering that current global economy is shaped while
various economies complicatedly affect each other

6. What is VAR
Impulse response function(1/2)
 “Impulse response function” is a function describing how a one-time shock (stress),
impulse, applied to a certain variable impacts on each variable and transmits.
 It allows use suitable for the purpose of stress test, as it can estimate for the future
how objective variables (e.g. Japanese long-term interest rate) are affected by a stress
event (e.g. one-time large drop of EU equity) considering correlation with other
variables and changed.
Impulse response： JPN Long Term Rate
Impulse response： EU Equity Price Index インパルス応答：JPN Long Term Rate
インパルス応答：EU Equity Price Index 0.00025
0
0.0002
-0.01
-0.02 One time shock
Transmission of shock 0.00015
0.0001
-0.03
0.00005
-0.04
0
-0.05
-0.00005
-0.06
-0.0001
-0.07
0 4 8 12 16 20 24 28 32 36 40
-0.08
0 4 8 12 16 20 24 28 32 36 40
Transmission to another economic variable
...

7. What is VAR
Impulse response function(2/2)
 Impulse response function is described as the following simple formula.
(Generalized impulse response function)
/
: Impulse response function after n period since the shock
(a shock of 1 standard deviation)
: row column element of variance/covariance matrix of the normal noise
: Coefficient matrix when inversely presenting model as model
: Variance/Covariance matrix of normal noise
: column vector of an unit matrix

8. Flow to Implement Stress Tests using VAR Model
Confirmation of goals of stress tests
→ What is “stress” for the company?
→ What “worst case” is assumed?
→ Consistency with measurement
methodology Stress test other than VAR
Select VAR
VAR modeling
Change in corporate
To prepare modeling in line
with goals of stress tests
• To select macro economic Calibration of factors
Managerial
value
indices
• To set trigger event Impulse response function judgment
• To set shocks
• To develop a satellite model Satellite model

9. Flow to Implement Stress Tests using VAR Model
Satellite Model
 Derivative model to incorporate impact of changes in macro economic
indices on corporate value
Example of VAR model Example of satellite model
Shocks on macro indices Real-world interest curve after the shock
• Short-term
+
Main
interest rate
Base
curve components of
yield curve
× Shock
• Long-term Credit risk spread after the shock
interest rate
Corporate
• Real GDP finance
model
× Shock → Change
in rating
• TOPIX Shocks on risk factors for other purposes
• CPI
=
Projection shock by
linear regression from ∆ ∆ ⋯
macro indices

10. Benefits to use VAR Model
1
Simplicity and
convincing to management
2
Compatibility with stress test
3
Linear characteristic

11. Benefits to use VAR Model
1
Simplicity and
convincing to
 Model is simple and clear, as it is basically
management expanded from autoregression model.
 Easy to explain the concept “correlation
2 between global economies and macro
Compatibility economies”.
with stress
test  Easy to graphically show as changes in well-
known economic variables.
3  It has experiences as a model (described later).
Linear
characteristic

12. Benefits to use VAR Model
1
Simplicity and
convincing to
 Easy to measure, as up/down movements after
management applying a stress is shown as an impulse
response function, an analytic formula
2  Able to measure the impact of stress for the
Compatibility future period
with stress
test  Impulse response function is not relative to
timing of occurrence of stress = Timing to put
3 stress can freely be set for a purpose
Linear
characteristic

13. Benefits to use VAR Model
 Characteristic as a linear model can be maintained, as it allows
matching as a linear model even against the past data showing non-
1
Simplicity and linear movement, when observing a single economic index.
convincing to – Additivity: &
management – Homogeneity：
 For example, simple (constant multiple) addition of impulse response
function can handle multiple stresses such as “occurrence of earthquake
2 disaster makes large decline in equity price and occurrence of sovereign
Compatibility shock abroad in the following year”.
with stress
Shock on price due to shock Shock on price due to shock
test on index X at t=0 on variable Y at t=4
Total shock on price
＋ ＝
3
Linear
characteristic  In contrast, acceptable change in corporate value can be reversely
calculated from multiple of standard deviation of a trigger event, which
is set as an early alert, and lead to management action if it goes beyond
the criteria.

14. Challenges to model VAR
 Too much observation data to gather
 Too many factors to determine before estimating parameters
– Determination of variables to use
– Whether any prior process is required (utilization of steps)
– Model lag
– And others
 Adjustment after estimation may be necessary
– Handling of a factor having poor fit (high p-value)
– Measures, when estimated value turns out to be unrealistic (such as negative
interest rate)
– And others
Here, Correct model ≠ Good model
Better to adjust and/or simplify depending on the goal of stress test

15. Experience of VAR Model
 Overseas central banks actively use VAR model to measure risks and evaluate
effect of economic and/or financial policies.
 Bank of Japan has been using VAR model as a stress test to check “robustness
of financial system to macro economic shock” since 2007 under “Financial
system report” published twice a year.
– The result of applying 5% probability shock simultaneously to real GDP and TOPIX
on VAR model using 5 variables of domestic economic indices is incorporated into
a satellite model (rating transition matrix, etc.) simulating Tier I ratio.
 While experience of private organizations using VAR model is not known in
detail, as their internal models are normally not disclosed, we know such
model is used at some of both insurance companies and reinsurance
companies.

16. GVAR Model
 VAR model may have concerns in accuracy and stability in estimating factors,
when the number of economic indices to incorporate increases as it increases
the number of factors to estimate significantly.
 A method to improve the accuracy of estimation has been considered by
developing and combining separate VAR model for each economy (referred as
VARX model). It is called GVAR model (Global Autoregression Model).
 European Central Bank seems especially active and issuing paper on GVAR
model. (as there are various economic indices of each EU member country?)

17. Image of Implementation
 MatLab has implemented modeling using "GVAR Toolbox 1.1" developed by L.
Vanessa Smith & Alessandro Galesi of Cambridge University.
 It models 7 economic indices variables of 33 countries using GVAR.
 Toolbox allows detailed selection of inclusion/non-inclusion or lag of variables
by country, of those results are automatically output in Excel files.
 Data accompanying Toolbox is used as is for this time and detailed conditions
are not considered specifically.
※ Results presented this time are just for illustration. Please pay attention in
using the data as its reasonableness is not fully considered.

18. Image of Implementation
Future estimate of economic indices (2010Q1 and thereafter)
JPY Real GDP JPN Long Term Rate
EU Real GDP US Long Term Rate
※ Results presented this time are just for illustration. Please pay attention in using the data as its reasonableness is not fully considered.

19. Image of Implementation
Projection of impact of EU equity shock on Japanese interest curve
Impulse response: EU Equity Price Index
インパルス応答：EU
Equity Price Index Impact of interest shock on major components
0
金利ショックの主成分への影響
0.025
-0.01 0.02
0.015
-0.02 One-time shock 0.01
-0.03 0.005
0
-0.04
-0.005
-0.05 -0.01
-0.015 パラレル
Parallel shift
-0.06
-0.02 Bend shift
ベンド
-0.07 -0.025
-0.08 -0.03
0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40
Yield curve
after shock
Impulse response: JPN Short Term Rate Impulse response: JPN Long Term Rate
by interest model
インパルス応答：JPN Short Term Rate インパルス応答：JPN Long Term Rate
0.0003 0.00025
0.0002
0.0002
0.00015
0.0001
0.0001
0
0.00005
-0.0001
0
-0.0002 -0.00005
-0.0003 -0.0001
0 4 8 12 16 20 24 28 32 36 40 0 4 8 12 16 20 24 28 32 36 40
※ Results presented this time are just for illustration. Please pay attention in using the data as its reasonableness is not fully considered.

20. Limitations and Disclosures
 Contents of the presentation is based on view of the presenter and
does not represent the employer of the presenter or MathWorks.
 Contents of the presentation does not represent formal opinion or
interpretation of the standards of practice as an actuary.
 Contents of the presentation have been developed to present general
information for sole purpose of education and does not intend for
completeness in terms of integrity or accuracy.
 Since it does not consider specific situation, users are advised to
consult with appropriate professionals before any decision making.
 Any of the presenter, the employer of the presenter or MathWorks shall
not be liable for any damages caused directly or indirectly relating to
the contents of the presentation.

21. Appendix：
Summary of Methods for Macro Stress Test
in ”Financial System Report” published by
Bank of Japan (BoJ)

22. Appendix: BoJ Macro Stress Test Models
Credit risk of bank lending + Equity risk of cross-shareholdings
Credit cost model
Real effective Financial situation of
foreign borrower Transition probability of
(ICR, cash-to-current
Credit cost
exchange rate debtor’s classification*
liabilities ratio)
5% probability Real GDP
shock Negative impact in line with lower growth rate
GDP deflator Nominal GDP Tier I Ratio
Equity valuation simulation
5% probability TOPIX Equity price Market Beta Equity valuation gain & loss
shock
Long-term Income simulation
lending interest Long-term
rate lending interest Lending spread Core business net income
rate
VAR model
Economic forecast of * = transition probability from rank m to n for company i (omitted m/n from formula)
private think tank ,
+ Nominal GDP increase ( ICR quick ratio)
,

23. Appendix: BoJ Macro Stress Test Models
Interest rising risk
3 types of interest rate rise * Lending interest rate at time = t （same formula in procurement interest rate）
・Parallel shift
（All term 1% up）
Trading interest model
・Steep-ize
（10-yr rate 1% up）
・Flat-ize Lending Lending
（Overnight rate 1% up） interest rate* interest
procurement Procurement
Interest
interest rate* interest
Stressed Bond interest
Tier I ratio
market yield
curve
Bond return Bond
Bond value valuation
Discount rate gain/loss
Bond valuation simulation
• In this Interest rising risk consideration, BoJ sets shifts of yield curve directly, not via VAR
model.
• On the contrary, as an illustration showed in the previous pages, yield curve shifts also
induced by macro economic stress via VAR model. We can synthesize the trigger events
into common economic stresses we used in the credit risk of bank lending and equity risk
of cross-shareholdings.

24. Appendix: BoJ Macro Stress Test Models
Market value loss risk of securities against shock in overseas market
TOPIX
S&P500 Stock price Fair value loss
Tier I ratio
decrease on stocks held
1% probability STOXX
shock Europe 600
Satellite model
VAR model
(daily return）
Japan gov.
Interest rate Fair value loss
US gov. Tier I ratio
increase on bonds held
1% probability Germany gov.
shock
Satellite model
VAR model
(10 yr bond yield)
• Use historical data during 1 year when the 3 variables became most correlated since 2000
respectively, and 1 year for time horizon.
（Stock：Aug. 2010 – Aug. 2011, Gov. bond：Oct. 2003 – Oct. 2004）

25. Appendix: BoJ Macro Stress Test Models
Other risks
 Other stress tests held in the report:
– “Foreign currency illiquidity risk”
：Assumes one-month malfunction of foreign currency swap market, repo
market and CD/CP market.
– “Loss enlargement risk due to interaction of financial capital market and
real economy”
：Assumes simultaneous shocks to STOXX Europe 600 and Germany
government bond yield and their remnants in the market for 3 years with
loss enlargement due to interaction of financial capital market and real
economy, using “Financial Macro-econometric Model (FMM)”