Portfolio Optimization with Drawdown Constraints

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Portfolio Optimization with Drawdown Constraints
January 29, 2000

• Alexei Chekhlov, TrendLogic Associates, Inc.
• Stanislav Uryasev Mikhail Zabarankin,
  University of Florida, ISE

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Introduction

• Losing clients accounts is equivalent to death
  of business
• Highly unlikely to hold an account which was in a
  drawdown for 2 years
• Highly unlikely to be permitted to have a 50
  drawdown
• Shutdown condition 20 drawdown
• Warning condition 15 drawdown
• Longest time to get out of a drawdown - 1 year.

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- uncompounded portfolio value at time t
- set of unknown weights
- drawdown function.

• Three Measures of Risk
• Maximum drawdown (MaxDD)
• Average drawdown (AvDD)
• Conditional drawdown-at-risk (CDaR)

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Limiting the risk

• MaxDD
• AvDD
• DVaR
• Combination

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Continuous Optimization Problems
MaxDD
AvDD
CDaR
technological constraints
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Discrete Optimization Problems
MaxDD
AvDD
CDaR
, (g)max0,g.
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Reward/Risk Ratios MaxDD
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Reward/Risk Ratios AvDD
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Table 1 MaxDD Solution
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Table 2 AvDD Solution
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Figure 1 MaxDD Efficient Frontier
Figure 2 AvDD Efficient Frontier
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Figure 3 Efficient Frontier as a function of
MaxDD
Figure 4 Efficient Frontier as a function of
AvDD
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Figure 5 MaxDDRatio as a function of MaxDD
Figure 6 AvDDRatio as a function of AvDD
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Underwater Curves MaxDD and AvDD
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Conclusions

• Introduced a one-parameter family of risk
  measures based on a notion of a drawdown
  (underwater) curve
• Mapped Portfolio Allocation problem into linear
  programming problems to be solved using efficient
  computer solvers
• Solved a particular real-life example on the
  basis of historical equity curves
• CDaR-generated solutions are more stable for
  practical weights allocation.