Introduction of VAR/GVAR Model as a Methodology to Develop Stress Test Scenarios for Market Risks

Motoharu DeiMilliman, Inc.

July 5, 2012

VAR = Vector Autoregression, GVAR = Global Vector Autoregression

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

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”

VAR = Vector Autoregression, GVAR = Global Vector Autoregression

I will introduce VAR/GVAR

as one of the technical solutions in introducing stress tests.

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.

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

・・・

To set it as a macro economic index (e.g. domestic and foreign equity indices, long- and short-term interest rates, price index)

: Time-series variable vector: Coefficient matrix

: Constant term vector: (Normal) Noise vector

What is VARImpulse 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.

Transmission of shock

-0.0001

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0

0.00005

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0.00015

0.0002

0.00025

0 4 8 12 16 20 24 28 32 36 40

インパルス応答：JPN Long Term RateImpulse response： JPN Long Term Rate

Transmission to another economic variable

...

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-0.07

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0

0 4 8 12 16 20 24 28 32 36 40

インパルス応答：EU Equity Price IndexImpulse response： EU Equity Price Index

One time shock

Impulse response function is described as the following simple formula.

What is VARImpulse response function(2/2)

(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

/

Flow to Implement Stress Tests using VAR Model

To prepare modeling in line with goals of stress tests

• To select macro economic indices

• To set trigger event

• To set shocks

• To develop a satellite model

Change in corporate

value

Managerial judgment

Stress test other than VAR

Select VAR

Confirmation of goals of stress tests

→ What is “stress” for the company?

→ What “worst case” is assumed?

→ Consistency with measurement methodology

Calibration of factors

Impulse response function

Satellite model

VAR modeling

Derivative model to incorporate impact of changes in macro economic indices on corporate valueExample of VAR model Example of satellite modelShocks on macro indices

• Short-term interest rate

• Long-term interest rate

• Real GDP

• TOPIX

• CPI

Real-world interest curve after the shock

Base curve

Main components of

yield curveShock+ ×

Credit risk spread after the shock

Corporate finance model

Shock Change in rating× →

Flow to Implement Stress Tests using VAR ModelSatellite Model

Shocks on risk factors for other purposes

Projection shock by linear regression from

macro indices= ∆ ∆ ⋯

Benefits to use VAR Model

Simplicity and

convincing to management

1

Compatibility with stress test

2

Linear characteristic

3

Benefits to use VAR Model

Model is simple and clear, as it is basically expanded from autoregression model.

Easy to explain the concept “correlation between global economies and macro economies”.

Easy to graphically show as changes in well-known economic variables.

It has experiences as a model (described later).

Simplicity and convincing to management

1

Compatibility with stress

test

2

Linear characteristic

3

Benefits to use VAR Model

Simplicity and convincing to management

1

Compatibility with stress

test

2

Linear characteristic

3

Easy to measure, as up/down movements after applying a stress is shown as an impulse response function, an analytic formula

Able to measure the impact of stress for the future period

Impulse response function is not relative to timing of occurrence of stress = Timing to put stress can freely be set for a purpose

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-linear movement, when observing a single economic index.– Additivity:

– Homogeneity：

Simplicity and convincing to management

1

Compatibility with stress

test

2

Linear characteristic

3

For example, simple (constant multiple) addition of impulse response function can handle multiple stresses such as “occurrence of earthquake disaster makes large decline in equity price and occurrence of sovereign shock abroad in the following year”.

&

＋ ＝

Shock on price due to shockon index X at t=0

Shock on price due to shockon variable Y at t=4

Total shock on price

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.

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 modelBetter to adjust and/or simplify depending on the goal of stress test

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.

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?)

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.

EU Real GDP

JPY Real GDP

※ Results presented this time are just for illustration. Please pay attention in using the data as its reasonableness is not fully considered.

Image of ImplementationFuture estimate of economic indices (2010Q1 and thereafter)

US Long Term Rate

JPN Long Term Rate

Image of ImplementationProjection of impact of EU equity shock on Japanese interest curve

※ Results presented this time are just for illustration. Please pay attention in using the data as its reasonableness is not fully considered.

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金利ショックの主成分への影響

パラレル

ベンド

Impact of interest shock on major components

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0 4 8 12 16 20 24 28 32 36 40

インパルス応答：EU Equity Price IndexImpulse response: EU Equity Price Index

One-time shock

-0.0001

-0.00005

0

0.00005

0.0001

0.00015

0.0002

0.00025

0 4 8 12 16 20 24 28 32 36 40

インパルス応答：JPN Long Term RateImpulse response: JPN Long Term Rate

-0.0003

-0.0002

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0

0.0001

0.0002

0.0003

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インパルス応答：JPN Short Term RateImpulse response: JPN Short Term Rate

Yield curve

after shock

by interest model

Parallel shift

Bend shift

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.

Appendix：

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

Appendix: BoJ Macro Stress Test ModelsCredit risk of bank lending + Equity risk of cross-shareholdings

* = transition probability from rank m to n for company i (omitted m/n from formula),

, + Nominal GDP increase ( ICR quick ratio)

Financial situation ofborrower

(ICR, cash-to-current liabilities ratio)

Nominal GDP

Equity price

Long-term lending interest

rate

Negative impact in line with lower growth rate

Transition probability of debtor’s classification* Credit cost

Market Beta Equity valuation gain & loss

Tier I RatioGDP deflator

TOPIX

Long-term lending interest

rate

Real effective foreign

exchange rate

Real GDP

VAR model

Economic forecast ofprivate think tank

Credit cost model

Equity valuation simulation

Income simulation

Lending spread Core business net income

5% probabilityshock

5% probabilityshock

Appendix: BoJ Macro Stress Test ModelsInterest rising risk

Lending interest rate*

Tier I ratio

• 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.

Stressed market yield

curve

procurement interest rate*

Bond return

Discount rate

Lending interest

Procurement interest

Bond interest

Bond valueBond

valuation gain/loss

Interest

Trading interest model

Bond valuation simulation

3 types of interest rate rise・Parallel shift

（All term 1% up）・Steep-ize

（10-yr rate 1% up）・Flat-ize

（Overnight rate 1% up）

* Lending interest rate at time = t （same formula in procurement interest rate）

Appendix: BoJ Macro Stress Test ModelsMarket value loss risk of securities against shock in overseas market

Tier I ratio

• 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）

TOPIX

S&P500

STOXX Europe 600

VAR model(daily return）

1% probability shock

Japan gov.

US gov.

Germany gov.

VAR model(10 yr bond yield)

Stock price decrease

Fair value loss on stocks held

Satellite model

Tier I ratioInterest rate increase

Fair value loss on bonds held

Satellite model1% probability shock

Appendix: BoJ Macro Stress Test ModelsOther 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)”