Value at risk

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Value at risk

• Anton Velkov Iliya Tsekov

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Risk Definition

• Uncertainty lack of certainty with
  understanding that there exists other
  possibilities of outcomes
• There is 90 chance that the market goes up
  tomorrow ? there is 10 uncertainty
• Risk State of uncertainty with understanding
  that at least one outcome involves a loss
• There is 10 chance that my investment will lose
  money today

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Value at Risk (VaR)

• We are X percent certain that we will not lose
  more than V dollars in time T.
• Function of confidence level X and time T

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Brief History

• Increasing need for risk management after the
  1987 market crash
• J.P. Morgan employees credited for developing VaR
• Known as the 415pm report
• RiskMetrics spinoff in 1994
• CreditMetrics and CorporateMetrics

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How to calculate VaR

• Historical Simulation
• Variance-Covariance Method
• Monte-Carlo Simulation

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Historical Simulation

• Forecasting VaR based on past data
• Example 1
• Calculate 1-day VaR at 99 confidence level for
  a portfolio with current value 23.5 million.
  You have 501 observations at your disposal.
• Figure out the market variables affecting the
  portfolio often exchange rates, equity prices,
  interest rates, etc.

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99 confidence level, interested in worst 1 ? we
take the 5th worst value as our VaR
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Example 2
-45.52 1-day 99 VaR
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How to calculate VaR

• Historical Simulation
• Variance-Covariance Method
• Monte-Carlo Simulation

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Variance Covariance Method

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Multi-Asset Case

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How to calculate VaR

• Historical Simulation
• Variance-Covariance Method
• Monte-Carlo Simulation

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Back Testing

• Integral part of any practical model
• Look back at the data and see how well it would
  have performed if it were implemented earlier
• Testing a 1-day 99 confidence VaR
• If over a given period we lost more money than
  the VaR estimate on about 1 of the days, then
  the model works
• If we lost more than VaR on 7-8 or 0.002 of the
  days, then the model might be corrupt

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Stress Testing

• Testing how the portfolio would have performed
  under some of the most extreme market moves
• Jan. 8th 1988 SP500 moved 6.8 st.dev
• Apr. 10th 1992 10-year bond yields moved by 7.7
  st.dev
• Oct. 19th 1987 SP500 moved 22.3 st.dev

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Artzner et al. (1998)

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Shortcomings of VaR

• Does not satisfy the subadditivity function of a
  coherent risk measure
• Does not put weight on anything else but the
  cutoff percentile the VaR
• Traders find cheating the VaR measure easy

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Low VaR, huge potential Loss
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Ascerbi Tasche (2001)

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Expected Shortfall vs. VaR

• Expected Shortfall passes all the coherent risk
  measure tests as presented by Artzner et al.
  (1998)
• Harder to back test when compared to VaR
• More difficult to interpret than VaR
• In spite of its weaknesses, VaR is more popular
  among both regulators and risk managers.

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Thank you

• Any questions?