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  • 1. Value at RiskJune 7, 2013Amir Khwajawww.clarusft.com
  • 2. Agenda The Need for VaR Definition of VaR Uses of VaR VaR Methods VaR - Historical Simulation Changes since the Financial Crises of 2008 Strengths and Weakness Summary
  • 3. The Need for VaR Different Asset Classes use their own measures Fixed Income – Duration Interest Rates – DV01 FX – Currency position Commodity – Number of contracts Equities – Number of shares Using these to compare risk in these portfolios is likecomparing apples and oranges An investor or owner needs a simple measure that can beused in a consistent way to compare risk between theseportfolios And with the ability to aggregate risk appropriately Value at Risk is that risk measure
  • 4. Definition of VaR How much could we lose over a specified holding periodwith a defined probability So if a portfolio has a VaR of $20 million We need to know the confidence level used to calculate We need to know the holding period (time horizon) Say confidence level is 99% and holding period is 5 days This then means “We would expect to lose $20 millionor more over a 5 day period, in 1 out of 100 businessdays” Note it does not tell us whether on that 1 day we couldlose $21 million or $200 million! So if we are relying only on VaR for the answer to thatthen we are going to be in trouble For that we need Tail Measures or Stress Testing
  • 5. Uses of VaR Made public by JP Morgan in 1994 with RiskMetrics Widely adopted in the industry very quickly after that Particularly for Derivatives where measures such as grossnotional or position in contracts units, are not thatinsightful Basel II Capital Accord for Market Risk – 1995 Internal Model Capital is VaR times a multiplier set for eachbank by its regulator as between 3 & 4 Banks report VaR in Annual Financial Statements – 1997 Internal 1d VaR and Regulatory 10d VaR Clearing Houses for Initial Margin – 1999 (LCH SwapClear) Margin from defaulting member used to cover the market riskloss for the period it takes to close-out the portfolio
  • 6. VaR Methods Three main methods Parametric (aka Variance-Covariance or Delta-Gamma) Historical Simulation Monte-Carlo Simulation Different assumptions, calculation steps, computeefficiency but similar numbers for standard portfolios The most common is Historical Simulation As easiest to understand Simple assumptions on distributions of returns So if for our $20 million VaR portfolio, we also said that wehad used 5 Years of history as well as 99% and 5d We would say that “given how the market has performedin the past 5 years” our VaR estimate is $20 million
  • 7. VaR - Historical Simulation It relies on choosing A historical period, e.g. 4 Years A holding period e.g. 5 days Generating daily holding period returns in this period Calculating the P&L impact on a portfolio by applyingthese returns to today Ordering the P&L outcomes by decreasing loss Interpolating for a chosen confidence level e.g. 99%
  • 8. VaR - Historical Simulation-80.00-60.00-40.00-20.000.0020.0040.0060.0080.00USD 5Y Swap Rate5d returns (bps)Sep08 to Sep1220Nov08 > -60bps13Dec10 > 20bps
  • 9. VaR - Historical Simulation Assume our portfolio has a PV01 of $1million Assume for simplicity that USD 5Y Swap is the only risk factor For a 1 bps rise in the 5Y Swap rate, our Profit will be $1m For a 1 bps fall in the 5Y Swap rate, our Loss will be $1m We can calculate the PL Series for our portfolio by multiplying thebps returns on each day by $1 million, which is shown below-80.00-60.00-40.00-20.000.0020.0040.0060.0080.00Profit LossSep08 to Sep1220Nov08 > $60m
  • 10. VaR - Historical Simulation This PL Series Has a PL value for each business day from 5 Sep 08 to 4 Sep 12 A total count of 1043 values Each corresponds to a specific scenario date, starting on 5 Sep 08 The first element represents the PL outcome of applying the 5-dayreturn shift between 1 Sep 08 and 5 Sep 08 to todays market dataand todays portfolio We call this the PL vector of the portfolio The first few elements of which are shown below-26.51-5.97-12.75-6.47-3.29-15.83-14.76-37.08-9.98-16.5522.7671.5509/05/200809/08/200809/09/200809/10/200809/11/200809/12/200809/15/200809/16/200809/17/200809/18/200809/19/200809/22/2008For these dates
  • 11. VaR - Historical Simulation The PL vector can then be re-ordered by decreasing loss Keeping a note of the scenario date and PL of each The first part of this is shown below123456789101111/20/200812/17/200810/21/200810/22/200811/21/200806/17/200908/14/200912/18/200810/06/200810/07/200809/16/2008-62.58-56.16-47.79-46.24-44.95-42.47-41.29-39.73-39.62-37.45-37.08For these datesRe-order by PL Now we can determine the VaR Which we will define as 99% or the loss of the 11th worst PL (We could define as 10th worst or interpolate between 10th and 11th) So VaR is $37.08m Occurs on the scenario date of 16-Sep-08, we call this the VaR Date This is the week of Lehman’s bankruptcy filingVaR Date
  • 12. VaR - Historical Simulation050100150200250-70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80Frequency5d PLsSep08 to Sep12Mean -1.21Standard Deviation 13.03Kurtosis 2.63Skewness 0.17Range 134.13Minimum -62.58Maximum 71.55VaR 99% -37.08Count 1043 A Histogram is a good way to view the PL vector Allocate each PL to a bin range Frequency is high for small PLs, giving the distribution below
  • 13. VaR - Historical Simulation Zooming in to the largest lossesMean -1.21Standard Deviation 13.03Kurtosis 2.63Skewness 0.17Range 134.13Minimum -62.58Maximum 71.55VaR 99% -37.08Count 104311th largest lossLargest lossExpected Shortfall
  • 14. Changes since the Financial Crises of 2008 Basell Capital Accord, introduced Stressed VaR So Trading Book Capital is the higher of Firm’s Multiplier *VaR or Multiplier * Stressed VaR Where Stressed VaR covers a period of Market Stress The Financial Crises of 2008 qualifies as a period of stressfor most major markets Wider use of Tail measures Expected Shortfall (ES) or Worst Case Loss (WCL) Renewed focus that Stress testing must be performed andthe definition and results of these discussed within the firmand with regulators Hence US & European Regulatory Stress Tests
  • 15. Strengths and Weaknesses Strengths Reduces risk to a single $ amount Mark to market based measure (not original notional) Compare risk of different asset class portfolios Aggregate risk across portfolios Widely used since 1994 Weaknesses Reduces risk to a single $ amount Assumptions may be complex Depends on market prices being observable and similarbehaviour to that observed in the past Long-tailed properties of financial markets Portfolio diversification is not there in a Crisis So correlation goes to 1
  • 16. Summary It is crucial to understand any assumptions For VaR, these are Method e.g. Historical Simulation Confidence level e.g. 99.7% Holding Period e.g. 5 days Historical Period Used e.g. 5 Years And not use just a single VaR measure Or indeed discard VaR (and replace with what?) So, in addition to VaR Use Tail Measures e.g. ES or WCL Stress Tests – Historical and Hypothetical Independent price verification Gross measures
  • 17. Contact DetailsOur LinkedIn Page: Clarus Financial TechnologyOur Website: www.clarusft.comMy contact details: amir@clarusft.com

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