STABLE NON-GAUSSIAN ASSET ALLOCATION:A COMPARISON WITH THE CLASSICAL GAUSSIAN APPROACH

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STABLE NON-GAUSSIAN ASSET ALLOCATIONA COMPARISON
WITH THE CLASSICAL GAUSSIAN APPROACH

• Yesim Tokat,
• Svetlozar Rachev, and
• Eduardo Schwartz

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I. Introduction

• Strategic investment planning
• Stochastic programming with and without decision
  rules
• Characteristics of financial and macro data
  heavy tails, time varying volatility and
  long-range dependence

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I. Introduction

• Generate economic scenarios under Gaussian and
  stable Paretian non-Gaussian assumptions
• Different allocations depending on the utility
  function and the risk aversion level of the agent
• very low or high risk aversion
• typical risk aversion

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II. Multistage Stochastic Programming with
Decision Rules

• Discretize time into n stages, and use a decision
  rule at each time period
• Boender et al. (1998)ALM model for pension funds
• randomly generate initial asset mixes
• simulate against generated scenarios
• evaluate downside risk and contribution rate

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III. Scenario GenerationDiscrete Time Series
Approach (Wilkie 1986, 1995)
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Continuous Time Approach (Mulvey 1996, Mulvey
and Thorlacius 1998)
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IV. Stable Distribution

• Financial returns Excess kurtosis found by Fama
  (1965) and Mandelbrot (1963,1967), Balke and
  Fomby (1994)
• Why stable Paretian distribution?
• Fat tails and high peak compared to Gaussian
• Generalized Central Limit Theorem
• Parameters index of stability, skewness
  parameter, location parameter,
• scale parameter,

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V. Model Setup

• Asset allocation
• generate initial asset allocations (a fixed mix)
• simulate future economic scenarios
• update asset allocation every month using fixed
  mix rule
• calculate the risk and reward
• choose initial mix that gives the best
  risk-reward combination

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Problem Formulation
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Reward measure

• mean final compound return
• where is the compound return of initial
  allocation i in periods 1 though T under scenario
  s.

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Risk measures

• mean absolute deviation of final compound return
• mean deviation of final compound return

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Utility functions

• where c is the coefficient of risk aversion
• power utility
• where ? is the coefficient of relative risk
  aversion

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Scenario Generation
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Scenario Generation

• cascade structure similar to Mulvey (1996)
• monthly data (1965-1999)
• Box-Jenkins methodology
• fit ARMA models to the financial variables
• model the residuals as Gaussian and stable
  Paretian

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Scenario Generation

• Simulation of future scenarios
• Generate normal and stable distributions for the
  residuals of each model
• generate T-bill, T-bond, inflation, stock
  dividend growth rate and stock dividend yield
  scenarios
• each variable has an innovation every month
• (T-bill and T-bond are dependent, others are
  independent)

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Normal and Stable Fit for Treasury Bill
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Normal and Stable Fit for Inflation
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Normal and Stable Fit for Dividend Growth
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Normal and Stable Fit for Dividend Yield
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Estimated Normal and Stable Parameters for the
Innovations
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Simulation

• Generate 512 possible economic scenarios for the
  next 3 quarters
• Repeat the scenario tree 99 times
• Compare simulated scenarios with historical
  averages
• No back-testing yet

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The Historical vs. Simulation Averages of
Economic Variables
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Optimal Allocations under Normal and Stable
Scenarios, T 3 quarters,
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Optimal Allocations under Normal and Stable
Scenarios, T 3 quarters
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Optimal Allocations under Normal and Stable
Scenarios, T 3 quarters
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VII. Conclusion

• Optimal asset allocation may be sensitive to the
  distributional assumption
• We need to model heavy tails more realistically
• Future work stochastic programming without
  decision rules