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Extreme Modeling of Equity Portfolios

Extreme Modeling of Equity Portfolios. Overview. Basic Ideas Methodology Results and Conclusions. Introduction. Framework for measuring tail-end risk. The relationship between market variables are different under stressed conditions.

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Extreme Modeling of Equity Portfolios

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  1. Extreme Modeling of Equity Portfolios

  2. Overview • Basic Ideas • Methodology • Results and Conclusions

  3. Introduction • Framework for measuring tail-end risk. • The relationship between market variables are different under stressed conditions. • All statistical measures derived solely from extreme values of the historical data. • Intuitive tool that is a complement to VaR.

  4. Framework • Makes no assumptions about the normality of asset returns. • Non-probabilistic approach to worst-case scenarios. There are no assumptions about the probability distribution of the size of the crash or its timing. • Prior Art: (1) Wilmott and Hua “CrashMetrics” (2) G. Stoder, “Quadratic Maximum Loss”, (3) Thomas Wilson - “Delta, Gamma Portfolio Optimisation”.

  5. Benchmark and Correlation • Relate the changes in an asset in a portfolio to the corresponding changes in a benchmark, an index or a combination of indices, during extreme movements. • For ENE’s Energy-Intensive portfolio, the following indices were chosen • OSX : Oil services Index. • XNG : Natural Gas Index. • XOI : Oil Index. • SPX : S&P 500 Index. • UTY : Utility Index. • NDX : Nasdaq100. • AOORD : Australian All Ordinaries. • TSE100 : Toronto Stock Exchange.

  6. Coefficient (contd.) • Relative magnitude of the moves are given by where Si is the ith asset, xj is the jth index and ki is the crash index. • Good assumption during extreme stress events. • Note, ki not the same as CAPM b, because the coefficient is obtained only during extreme events: Haliburton return * * RHAL = .9654ROSX R2 = .954 * * * * * * OSX (Oil Services) return * * * *

  7. Change in Portfolio Value • If p is the value of the portfolio, then portfolio change Dp is given by where nSi is the ith stock, and nCi is the ith option. • The change in option value DC is given by • di and Gij are the delta and gamma terms. We neglect cross-gamma for now.

  8. Change in Portfolio Value (contd.) • Using the expression for , then portfolio change Dp is given by (for a single index: I use 6 indices) • Here y is a variable = return of the index. • If 6-indices are used I obtain a quadratic form with 6 variables. It looks like • Where y is a vector of index returns (variable), B is a diagonal matrix.

  9. Portfolio Optimization: Worst-case • The worst-case scenario optimization problem is then given by • Here y- and y+ are vectors of constraints for the range on the index returns. I have chosen historical highest and lowest for the index constraints. • Conjugate Gradient Algorithm used to solve above problem.

  10. Data Details • Reuter’s and Bloomberg utilized to obtain all the data. • 15 years for historical data used for all equities under OSX, XOI, SPX,AOORD. 12 years of data used for UTY and XNG, while 10 years of data used used for NDX. • Current VaR model interface used to obtain prices, deltas and Gammas.

  11. Results • Effective_Date = 24th August 2000 • Value@Risk = $1.21MM • Worst-Scale Loss = $5.8MM

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