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**Value-at-Risk: A Risk Estimating Tool for Management**

February 24, 2000 David Dudley Federal Reserve Bank of New York

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**Overview Brief History of Market Risk Measurement Methods**

Implementation of VaR Applications of VaR Unique Risks Capture Buy vs. Build Issues VaR Validation and Backtesting

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**History of Market Risk Measurement Methods**

Market Risk is the possibility of loss related to uncertain movements in market risk factors such as FX, interest rates, commodities and equities. Another factor for consideration is the determination of the magnitude of the losses over a given holding period or risk liquidation/mitigation period. In the beginning, there was reliance upon notional values but this provided no basis for comparison across instruments and or maturity periods. Sensitivity approaches were developed to measure the changes in prices of single instruments or factors across maturities but still full comparison on a portfolio basis was not meaningfully accomplished.

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**VaR: A Statistical Approach**

Given that the previous two “methods” failed to forecast “future” movements in markets, a statistical basis was needed to bridge this gap. Probability distributions, which relates magnitudes of all possible market value changes to their probabilities, is the central notion of VaR. VaR’s advantages are: Measures risks in terms of potential losses. Relates the losses to its probability (confidence intervals). Measures risks in all products using the same “units.” Allows the aggregation of risks of various positions into a single portfolio risk measure. Combines position factors - size,structure, volatility and holding period.

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Implementation of VaR Risk is measured by the “worst loss” for a given level of confidence under a normal distribution (e.g. 2 SD gives 1 tail of 97.5%). Estimate the probability distributions for all relevant market risk factors (historical data). Derive the probability distribution of market values for each instrument within the portfolio. Aggregate the probability distributions of changes in market values for individual instruments into the probability distribution for total portfolio to measure portfolio VaR. Calculate VaR.

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Market Factors Estimating the probability distributions of relevant market factors (e.g. distribution of bond yield changes with related probabilities) involves the usage of historical market data. A key starting point is the assumed holding or risk liquidation/mitigation period of the modeled instrument. One major assumption is that distributions are “stationary” in the near term (e.g. the past predicts the future). A major parameter is the length of the lookback period (e.g. amount of historical data). This last point depends heavily upon the availability of data. If the desired effect is not possible, “proxies” could be used but carry additional modeling risks.

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Market Values The calculated distribution of changes in market factors are now applied to positions to evaluate the distribution of changes in market values (e.g. changes in the price of bonds for changes in yields). Alternative Methods to calculate Market Value Distributions: Sensitivity based approach for options to include all changes in risk factors; Interpolation between pre-calculated market values and current market value of instrument; Full valuation - using pricing algorithms of modeled instruments (swaps and other derivatives).

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Portfolio VaR Combine the distribution of changes in market values for instruments into a single distribution for the entire portfolio to measure the portfolio VaR. Depending upon the VaR methodology utilized by the institution this may be a relatively easy exercise or may not be determined with statistical meaningfulness. Keep in mind that VaR is more “Art” than “Science” and there is “no truth” to a single number. VaR determines a range of outcomes. For a 97.5% confidence interval determining for example a VaR of $20 million, this says that 2.5% of the time period or 1 year (250 trading days) losses will exceed $20 million on 7.5 days.

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**VaR Methods Closed Form VaR or Variance/Covariance**

Historical Simulation Approach Monte Carlo Modeling

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**Closed Form VaR or Var/Cov.**

The closed for model assumes that Portfolio probability is normally distributed Profitability is a linear relationship with applicable risk factors With these assumptions the VaR can be calculated directly from the volatilities and correlations of the applicable risk factors. Portfolios which may fit this approach include equities, spot & forward FX transactions, commodities and short-term debt instruments. Options, structured notes and mortgage-backed instruments which all have gamma/convexity (curvature in price changes for different risk factor changes) are not modeled well under this method.

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**Historical Simulation Approach**

Price scenarios are drawn directly from historical price movements as opposed to risk factor changes. The 1 day VaR for a lookback period of 1 year would include: Collect for how applicable key factors performed over the past 250 trading days. For each day, determine the percent change in each of the key factors to produce day scenarios for the performance of these key factors; Revalue the current portfolio under each of the 250 scenarios, determining for each one what would be the portfolio’s P/L if that scenario were to occur over the next 24 hours; Form a “histogram” (distribution) from which the VaR can be estimated for the desired confidence interval.

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Monte Carlo Methods This is a methodology for estimating risk for complex portfolios. Whereas the closed-form approach produces “exact” results for VaR of simple portfolios, it can produce erroneous results for portfolios exhibiting non-linear relationships. Monte Carlo uses statistical techniques to randomly construct a histogram of possible P/L for a portfolio over a specified time horizon. Through the generation of “scenarios” the potential outcomes are determined to which a confidence interval is applied. This method requires significant computer resources to generate the appropriate number of according to current academic and industry norms. Timeliness of reporting can be an issue here.

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Applications of VaR VaR enhances the risk management and reporting capabilities within the institution: Reporting and disclosure of risk Measurement an monitoring of market risks Controlling the risks through limits Can be used to identify optimized hedging Allocation of market risk capital within the institution Measure performance of revenue generating activities on a risk-adjusted basis.

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Backtesting To determine the ongoing adequacy of the VaR model we need to examine its performance. This process involves the review of expected P/L (VaR calculation) with actual P/L. Actual P/L must be based upon the review of the P/L of end of day positions and elimination of intraday trading P/L and any associated fee income (structuring fees) received by the trading group. Separation of P/L or P/L attribution is a key element here and is also a useful management tool for evaluating revenues. P/L systems and controllers need to amend traditional systems to accomplish this. Determine acceptable ranges of P/L and comparisons with VaR.

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The End

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CHAPTER 10 Market Risk Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.McGraw-Hill/Irwin.

CHAPTER 10 Market Risk Copyright © 2011 by The McGraw-Hill Companies, Inc. All Rights Reserved.McGraw-Hill/Irwin.

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