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Scenario-Based Markowitz Portfolio Optimization With Discontinuous Risk

Scenario-Based Markowitz Portfolio Optimization With Discontinuous Risk. Michael Aiello Polytechnic University. Traditional Capital Asset Pricing Model. Assumes that asset returns are lognormally distributed, random variables.

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Scenario-Based Markowitz Portfolio Optimization With Discontinuous Risk

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  1. Scenario-Based Markowitz Portfolio Optimization With Discontinuous Risk Michael Aiello Polytechnic University

  2. Traditional Capital Asset Pricing Model • Assumes that asset returns are lognormally distributed, random variables. • However, historical period returns show more spikes of more than 2 standard deviations from the mean than expected.

  3. Developing the Model • From a “full history” view, prices do not act as lognormally distributed random variables at all times • Returns may act more like lognormal variables during periods between spikes • If data from different “areas” or “scenarios” is defined within the weighing model, a more appropriate weighting may be achieved.

  4. Approach • Take into consideration the number of “spikes” or “jumps” as an additional risk factor. The risk model should shift as a reaction to these known jumps. • Return scenarios and correlations should be non-constant in order to best represent different markets (i.e. bull vs bear)

  5. Efficient Frontier With Scenario Consideration • Efficient frontiers for weekly returns during bull and bear markets differ significantly for same set of stocks

  6. Efficient Frontier Considering Equally Likely Scenario Returns with Bull Covariances

  7. Efficient Frontier Considering Equally Likely Scenario Returns with Bear Covariances

  8. Comparison of Returns and “Jumps” Bull Year(1999)

  9. Comparison of Returns and “Jumps” Bear Year(2002)

  10. Modification to traditional model Original Model Modified Model, adds Jumps/B percentage points to covariance risk for each jump outside 2 standard deviations. B is “Jump Influence” should be 10-30

  11. Note: The models examined 17 stocks. ADM, APC, BSC, CHIR, EZPW, FRX, FTO, GENZ, LEH, MSFT, SUN, TYC, UNH, UTX are not shown as they were assigned weights of 0 in both models. Resulting Mix Traditional Mix at 0.84% Weekly Return and 0.12 Risk Risk with Jumps Mix at 0.80% Weekly Return and 0.09 Risk

  12. $10000 investment results • Purchased suggested mix on Jan 1 2000 and sold on Dec 31 2000 • Jump Model • Start $9979.34, End $13256.00 • Return 32.83% • Traditional Model • Start $9941.64, End $13803.91 • Return 38.84%

  13. Future Work and Concerns • Larger data sets • Strict definition of bull and bear markets • Monte Carlo simulations to select B and the market Bull/Bear probabilities • In the end, selection of inclusion should be up to the customer. • Results in a “smoother” value curve which is more appealing to investors.

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