ScenarioBased Markowitz Portfolio Optimization With Discontinuous Risk

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Scenario-Based Markowitz Portfolio Optimization
With Discontinuous Risk

• Michael Aiello
• Polytechnic University

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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.

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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.

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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)

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Efficient Frontier With Scenario Consideration

• Efficient frontiers for weekly returns during
  bull and bear markets differ significantly for
  same set of stocks

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Efficient Frontier Considering Equally Likely
Scenario Returns with Bull Covariances
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Efficient Frontier Considering Equally Likely
Scenario Returns with Bear Covariances
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Comparison of Returns and Jumps Bull Year(1999)
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Comparison of Returns and Jumps Bear Year(2002)
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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
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(No Transcript)
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Resulting Mix
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.
Traditional Mix at 0.84 Weekly Return and 0.12
Risk
Risk with Jumps Mix at 0.80 Weekly Return and
0.09 Risk
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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

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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.