Introduction of VAR/GVAR Model as a Methodology to Develop Stress Test Scenarios for Market Risks
Introduction of VAR/GVAR Model as aMethodology to Develop Stress TestScenarios for Market RisksMotoharu DeiMilliman, Inc.July 5, 2012VAR = 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 AppendixVAR = 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”
I will introduce VAR/GVAR as one of the technical solutions in introducing stress tests.VAR = Vector Autoregression, GVAR = Global Vector Autoregression
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
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 ...
What is VARImpulse 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
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 corporateTo prepare modeling in linewith 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
Flow to Implement Stress Tests using VAR ModelSatellite 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
Benefits to use VAR Model 1 Simplicity and convincing to management 2 Compatibility with stress test 3 Linear characteristic
Benefits to use VAR Model1Simplicity andconvincing to Model is simple and clear, as it is basically management expanded from autoregression model. Easy to explain the concept “correlation2 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). Linearcharacteristic
Benefits to use VAR Model1Simplicity andconvincing to Easy to measure, as up/down movements after management applying a stress is shown as an impulse response function, an analytic formula2 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 put3 stress can freely be set for a purpose Linearcharacteristic
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-1Simplicity 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 earthquake2 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 Linearcharacteristic 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 model Better 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.
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.
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.
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Appendix：Summary of Methods for Macro Stress Testin ”Financial System Report” published byBank of Japan (BoJ)
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 simulation5% 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) ,
Appendix: BoJ Macro Stress Test ModelsInterest rising risk3 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.
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 held1% 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 held1% 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）
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)”