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## Value-at-Risk (VaR)

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**Value-at-Risk (VaR)**Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html**Risk**• Business Risk • Financial Risk • market risk • credit risk • liquidity risk • Operational Risk • Legal Risk FRM5-VaR**How much can we lose?**Everything correct, but useless answer. How much can we lose realistically? FRM5-VaR**duration, convexity**volatility delta, gamma, vega rating target zone What is the current Risk? • Bonds • Stocks • Options • Credit • Forex • Total ? FRM5-VaR**Standard Approach**FRM5-VaR**Modern Approach**Financial Institution FRM5-VaR**Definition**VaR is defined as the predicted worst-case loss at a specific confidence level (e.g. 99%) over a certain period of time. FRM5-VaR**VaR1%**1% Profit/Loss VaR FRM5-VaR**VaR**1% Meaning of VaR A portfolio manager has a daily VaR equal $1M at 99% confidence level. This means that there is only one chance in 100 that a daily loss bigger than $1M occurs, under normal market conditions. FRM5-VaR**Main Ideas**• A few well known risk factors • Historical data + economic views • Diversification effects • Testability • Easy to communicate FRM5-VaR**History of VaR**• 80’s - major US banks - proprietary • 93 G-30 recommendations • 94 - RiskMetrics by J.P.Morgan • 98 - Basel • SEC, FSA, ISDA, pension funds, dealers • Widely used and misused! FRM5-VaR**Current position**Market data Risk Mapping Valuation Value-at-Risk Reporting and Risk Management Risk Management Structure FRM5-VaR**Value**dollar Interest Rate interest rates and dollar are NOT independent FRM5-VaR**Risk Measuring Programs**• CATS, CARMA $400K/yr • Algorithmics, Risk Watch >$1M • Infinity >$1M • J.P. Morgan, FourFifteen $25K/yr • FEA, Outlook $18K • Reuters, Sailfish ? • Theoretics, TARGA $75K • Bankers Trust, RAROC $50K/run • INSSINC, Orchestra $25-75K FRM5-VaR**Why Capital Requirements**• Government Protection • Debt financing • In Israel this is not binding! • But it will be!! • Traded assets constitute a small portion • 339-9, 339-10 defines who must use VaR FRM5-VaR**Unifying Approach**• One number • Based on Statistics • Portfolio Theory • Verification • Widely Accepted • Easy Comparison FRM5-VaR**Supervision factor, 3 k 4**Capital Requirements Minimal required capital FRM5-VaR**Qualitative Requirements**• An independent risk management unit • Board of directors involvement • Internal model as an integral part • Internal controller and risk model • Backtesting • Stress test FRM5-VaR**Quantitative Requirements**• 99% confidence interval • 10 business days horizon • At least one year of historic data • Data base revised at least every quarter • All types of risk exposure • Derivatives FRM5-VaR**Types of Assets and Risks**• Real projects - cashflow versus financing • Fixed Income • Optionality • Credit exposure • Legal, operational, authorities FRM5-VaR**Risk Factors**There are many bonds, stocks and currencies. The idea is to choose a small set of relevant economic factors and to map everything on these factors. • Exchange rates • Interest rates (for each maturity and indexation) • Spreads • Stock indices FRM5-VaR**How to measure VaR**• Historical Simulations • Variance-Covariance • Monte Carlo • Analytical Methods • Parametric versus non-parametric approaches FRM5-VaR**Historical Simulations**• Fix current portfolio. • Pretend that market changes are similar to those observed in the past. • Calculate P&L (profit-loss). • Find the lowest quantile. FRM5-VaR**Example**Assume we have $1 and our main currency is SHEKEL. Today $1=4.30. Historical data: 4.00 4.20 4.20 4.10 4.15 P&L 0.215 0 -0.112 0.052 4.30*4.20/4.00 = 4.515 4.30*4.20/4.20 = 4.30 4.30*4.10/4.20 = 4.198 4.30*4.15/4.10 = 4.352 FRM5-VaR**USD NIS**2000 100 -120 2001 200 100 2002 -300 -20 2003 20 30 today FRM5-VaR**today**USD: +1% +1% +1% +1% NIS: +1% 0% -1% -1% Changes in IR FRM5-VaR**1% of worst cases**Returns year FRM5-VaR**VaR1%**1% Profit/Loss VaR FRM5-VaR**Variance Covariance**• Means and covariances of market factors • Mean and standard deviation of the portfolio • Delta or Delta-Gamma approximation • VaR1%= P – 2.33 P • Based on the normality assumption! FRM5-VaR**1%**2.33 Variance-Covariance -2.33 FRM5-VaR**Monte Carlo**FRM5-VaR**Monte Carlo**• Distribution of market factors • Simulation of a large number of events • P&L for each scenario • Order the results • VaR = lowest quantile FRM5-VaR**Monte Carlo Simulation**FRM5-VaR**Weights**Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential. See RiskMetrics Technical Document for details. FRM5-VaR**Stock Portfolio**• Single risk factor or multiple factors • Degree of diversification • Tracking error • Rare events FRM5-VaR**Bond Portfolio**• Duration • Convexity • Partial duration • Key rate duration • OAS, OAD • Principal component analysis FRM5-VaR**Options and other derivatives**• Greeks • Full valuation • Credit and legal aspects • Collateral as a cushion • Hedging strategies • Liquidity aspects FRM5-VaR**Credit Portfolio**• rating, scoring • credit derivatives • reinsurance • probability of default • recovery ratio FRM5-VaR**Reporting**Division of VaR by business units, areas of activity, counterparty, currency. Performance measurement - RAROC (Risk Adjusted Return On Capital). FRM5-VaR**How VaR is used**• Internal Risk Management • Reporting • Regulators FRM5-VaR**Backtesting**Verification of Risk Management models. Comparison if the model’s forecast VaR with the actual outcome - P&L. Exception occurs when actual loss exceeds VaR. After exception - explanation and action. FRM5-VaR**Backtesting**OK increasing k intervention Green zone - up to 4 exceptions Yellow zone - 5-9 exceptions Red zone - 10 exceptions or more FRM5-VaR**Stress**Designed to estimate potential losses in abnormal markets. Extreme events Fat tails Central questions: How much we can lose in a certain scenario? What event could cause a big loss? FRM5-VaR**Risk Management**• Risk measurement • Reporting to board • Limits monitoring • Diversification, reinsurance • Vetting • Reporting to regulators • Decision making based on risk FRM5-VaR**Tool, not rule!**FRM5-VaR**pluto.mscc.huji.ac.il/~mswiener/**Risk Management resources • Useful Internet sites • Regulators • Insurance Companies • Risk Management in SEC reports FRM5-VaR**How to hedge financial risk?**• Static hedge Forwards agreements that fix the price Futures Options static hedge • Dynamic delta or vega hedge, with a variable amount of options held. It is applicable if there is a very liquid market and low transaction costs. FRM5-VaR**RMG**• http://www.riskmetrics.com/ • http://www.pictureofrisk.com/ • http://www.riskmetrics.com/rm/splash.html • rmgaccess FRM5-VaR**Who manages risk?**Nike Sony Dell Computers Philip Morris Ford Motor AIG General Re Swiss Re Aetna Zurich Citibank Bank of England CIBC J. P. Morgan Bankers Trust FRM5-VaR**Consulting**• Oliver, Wyman and Co. • Willis Corroon • Richard Scora • Ernst and Young • Enterprise Advisors • Kamakura FRM5-VaR