Introduction of VAR/GVAR Model as a Methodology to Develop Stress Test Scenarios for Market Risks
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Transcript of Introduction of VAR/GVAR Model as a Methodology to Develop Stress Test Scenarios for Market Risks
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
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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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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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インパルス応答:EU Equity Price IndexImpulse response: EU Equity Price Index
One-time shock
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インパルス応答:JPN Long Term RateImpulse response: JPN Long Term Rate
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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)”