Optimal Rebalancing Strategy for Pension Plans A Presentation to State Street Associates 15.451 Financial Engineering Proseminar MIT Sloan School of Management November 18, 2004

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Optimal Rebalancing Strategy for Pension PlansA
Presentation to State Street Associates15.451
Financial Engineering ProseminarMIT Sloan School
of ManagementNovember 18, 2004
Marius Albota Josh Grover Tom Schouwenaars Walter Sun
Li-Wei Chen Ayres Fan Ed Freyfogle Josh Grover Tom Schouwenaars Walter Sun
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Problem Summary

• Managers create portfolios comprised of various
  assets
• The market fluctuates, asset proportions shift
• Given that there are transaction costs, when
  should portfolio managers rebalance their
  portfolios?
• Most managers currently re-adjust either on
• a calendar basis (once a week, month, year)
• when one asset strays from optimal (/- 5)

Both of these methods are arbitrary and
suboptimal.
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Why is this problem important?

• An optimal rebalancing strategy would give a firm
  a measurable advantage in the marketplace
• Providing rebalancing services could be a
  significant new revenue stream for State Street

Getting this right would be worth lots (and we
mean lots) of money
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Presentation Outline

• Simple Example
• Our Solution
• Methodology
• Two Asset Model
• Multi-Asset Model
• Sensitivity Analysis
• Conclusion
• Future Research

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A Simple Example

• On Aug. 15 your portfolio was 50 invested in
  Nasdaq (QQQ)
• You go on a three month, round-the-world trip
• On Nov. 15 you waltz into the office, and realize
  your investment went up!!!

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A Simple Example (cont.)

• Sadly, the other 50 of your portfolio was
  invested in a long term bond fund (PFGAX)
• Long term bonds have underperformed recently

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A Simple Example (cont.)

• Your portfolio is now unbalanced.
• Should you rebalance now?
• When should you have rebalanced?
• What if the act of trading costs you 40 bps?
• 60 bps? or a flat fee per trade?
• Now imagine if you had many different assets, of
  all different types!!!
• What about taxes?

When and how to rebalance is complicated.
Transaction costs make it much more difficult.
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Our Solution

• In theory when to rebalance is easy

Rebalance when the costs of being suboptimal
exceed the transaction costs

• In practice the transaction cost is known
  (assuming no price impact).
• It is difficult to know the benefit of
  rebalancing.

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When to rebalance depends on three costs

• Cost of trading
• Cost of not being optimal this period
• Expected future costs of our current actions

The cost of not being optimal (now and in the
future) depends on your utility function
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Utility Functions

• Quantify risk preference
• Assume three possible utilities

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Certainty Equivalents

• Given a risky portfolio of assets, there exists a
  risk-free return rCE (certainty equivalent) that
  the investor will be indifferent to.
• Example 50 US Equity 50 Fixed-Income 5
  risk-free annually
• Quantifies sub-optimality in dollar amounts
• Example Given a 10 billion portfolio.
• The optimal portfolio xopt is equivalent to 50
  bps per month
• A sub-optimal portfolio xsub is equivalent to 48
  bps per month
• On this portfolio, that difference amounts to 2
  million per month

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Dynamic Programming - Example

• Given up to three rolls of a fair six-sided die
• Payout is 100 ? (result of your final roll)
• Find optimal strategy to maximize expected payout
• Solution
• Work backwards to determine optimal policy

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Dynamic Programming

• Examine costs rather than benefit
• Jt(wt) is the cost-to-go at time t given
  portfolio wt

Current period tracking error
Cost of Trading
Expected future costs

• Trade to wt1 (optimal policy)
• When wt1 wt, no trading occurs

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Data and Assumptions

• Given monthly returns for 8 asset classes and
  table of expected returns
• Used 5 asset model due to
• computational complexity
• lack of diversification in computed optimal
  portfolio

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Optimal Portfolios

• Calculated efficient frontier from means and
  covariances
• Performed mean-variance optimization to find the
  optimal portfolio on efficient frontier for each
  utility

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Two Asset Model

• Demonstrate method first on simple two asset
  model
• US Equity 7.06, Private Equity 14.13 (2
  risk-free bond)
• 10 year (120 period) simulation

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Two Asset Model
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Multi-Asset Model

• We construct the optimal portfolio from 5 of the
  8 assets
• Some assets were highly correlated with others,
  other were dominated
• US Equity, Developed Markets, Emerging Markets,
  Private Equity, Hedge Funds
• Ran 10,000 iteration Monte Carlo simulation over
  10 year period for all three utility functions

Quadratic Utility
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Simulation Results

• On average, with a 10 BN portfolio, our strategy
  will
• Give up 700 K in expected risk-adjusted return
• Save 3.5 MM in transaction costs

Netting 2.8 MM in savings!!!
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Simulation Results (cont.)
2.8 MM in savings!!!
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Sensitivity US Equity Returns
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Sensitivity Correlation
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Sensitivity US Equity Standard Deviation
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Conclusions

• Portfolio rebalancing theory is quite
  basicrebalance when the benefits exceed the
  transaction costs
• However, the calculation proves quite difficult
• The more assets involved, the harder it is to
  solve
• Our DP method outperformed all other methods
  across several utility functions

Use dynamic programming to save money
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Possibilities for Further Analysis

• Variable transaction cost functions
• Different utility functions
• Varying assumptions that could be challenged
• Tax implications
• Time to rebalance gt 0
• Impact of short sales

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ThanksSebastien Page, VP State Street Mark
Kritzman, Windham Capital Management
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Questions?