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Practical Problems with Building Fixed-Income VAR Models Rick Klotz Managing Director Global Head of Market Risk Management Greenwich NatWest

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Value At Risk: Definition The value at risk (VAR) of a portfolio is the loss in value in the portfolio that can be expected over a given period of time (e.g., 1-Day) with a probability not exceeding a given number (e.g., 5%). Probability (Portfolio Loss < - VAR) = K K = Given Probability

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Visualizing VAR: An Example A one day VAR of $10mm using a probability of 5% means that there is a 5% chance that the portfolio could lose more than $10mm in the next trading day.

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VAR and Capital Requirements For Banks Regulatory Capital = Market Risk Capital + Specific Risk Capital + Counterparty Risk Capital Market Risk Capital = Max [Ave. of 10-Day 99% VAR x Multiplier, yesterdays 10-Day 99% VAR]

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Back-Testing A VAR Model Calculate 1-Day 95% VAR for a (changing) portfolio each day for some substantial period of time (e.g., 100 Days) Compare the P/L on the succeeding trading day with the previous close of business days VAR Count the number of times the loss exceeds the VAR

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The Need For VAR Model Accuracy If the VAR is systematically too low, the model is underestimating the risk and you tend to have too many occasions where the loss in the portfolio exceeds the VAR. This can lead to an increase in the multiplier for the capital calculation. If the VAR is systematically too high, the model is over estimating the risk and your regulatory capital charge will be too high

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Building A VAR Model: Estimating the Change in the Value of the Portfolio Estimate the change in the value of the portfolio P, as a function of the change in the value of risk factors,..., (e.g.,, may be the change in 1-year U.S. interest rates, may be the change in 2- year U.S. interest rates, etc.). Example:

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Building A VAR Model: Basic Methodologies 1) Variance/Covariance Method - Use historical variances and covariances of risk factors,, to estimate how large 1.645 (for 5%) is for the distribution of.

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Building A VAR Model: Basic Methodologies 2) Historical Simulation Method - Take an historical period, say the last 501 trading days, and calculate Order from highest to lowest and take the 475th as the VAR

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Building A VAR Model: Basic Methodologies 3) Monte Carlo Simulation Method - Simulate a set of 500 (for example) by choosing for risk factors ( can be historical or implied from options, are usually historical). Order the from highest to lowest and take the 475th as the VAR.

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Model Specific Issues

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General Challenges for VAR Models Obtaining Good Historical Data Finding a complete set of risk factors - fixed income VAR models generally miss bond specific information (e.g., issuer specific risk) How to weight historical data to accurately determine a 1-day VAR.

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Obtaining Good Historical Data Poor Data – Even actively traded markets can have noisy historical data – Less actively traded markets can pose a significant challenge to finding clean historical data – Historical data can be misleading if a market is maturing over that period Missing Data – It may be difficult to find historical data in relatively new (e.g., U.K. Asset Backeds) or inactive markets (e.g., inverse I.O.s) Asynchronous Data – The data for risk factors that are traded against each other (e.g., Mortgages and Treasuries, Futures and Cash Securities, etc.) must reflect simultaneous closes.

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Finding a Complete Set of Risk Factors Fixed Income VAR models generally miss bond (even market) specific information –Coupon to coupon trades –Basis trades –Issuer specific risk –Some market specific risks (e.g., U.K. Asset Backeds) Some risk factors are mapped to risk factors that have adequate historical data but may not be good proxies

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How Should Historical Data Be Weighted To Calculate a 1-Day VAR? Regulators require that you use at least one year of historical data An option trader buying or selling a 1-Day option would give very little weight to old data

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1 Value at Risk Chapter 20. 2 The Question Being Asked in VaR “What loss level is such that we are X % confident it will not be exceeded in N business.

1 Value at Risk Chapter 20. 2 The Question Being Asked in VaR “What loss level is such that we are X % confident it will not be exceeded in N business.

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