Optimal Rebalancing Strategy for
Institutional Portfolios
Walter Sun
Joint work with Ayres Fan, Li-Wei Chen,
Tom Schouwenaars, Marius Albota,
Ed Freyfogle, Josh Grover
QWAFAFEW - Boston Meeting
April 12, 2005
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Problem Summary
• Managers create portfolios comprised of various assets & asset classes
• 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
• Optimal rebalancing can reduce the amount of trading
The ‘correct’ strategy can reduce costs.
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Presentation Outline
• Simple Example
• Our Solution
• Two Asset Model
• Multi-Asset Model
• Sensitivity Analysis
• Conclusion
• Future Research
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Example
• On Aug. 15, 2004, a portfolio was equal-weighted between the
Nasdaq 100 ETF (QQQQ) and a long-term bond fund (PFGAX).
• On Nov. 15, 2004, the portfolio is no longer equal-weighted, as
QQQQ (red) has gained 16.5% while PFGAX (blue) has increased
2%; so QQQQ now represents 53% of the portfolio.
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Example
• Your portfolio is now unbalanced.
• Should you rebalance now, or should you have rebalanced earlier?
• How much should it depend on your exact trading costs (40bps,
60bps, or flat fee)?
When and how to optimally rebalance is complicated.
Transaction costs make it much more difficult.
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Our Solution
• In theory, the decision rule is simple:
Rebalance when the costs of being suboptimal exceed the
transaction costs
• In practice the transaction cost is known (assuming
no price impact), but the cost of suboptimality is not.
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When to rebalance depends on three costs:
1. Cost of trading
2. Cost of not being optimal this period
3. 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
• J2(r2) – expected benefit at time 2, given roll of r2
• J2(r2) = max( r2, E(J3(r3)) ) = max( r2, 3.5 )
Roll
• J1(r1) = max( r1, E(J2(r2)) )
Roll
r1
r3
r2
Accept if r2>3.5
Accept if r1>E(J2(r2))
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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
•Trade to wt+1 (optimal policy)
–When wt+1 = wt, no trading occurs
Expected future
tracking error
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Data and Assumptions
• Given annual returns for 5 asset
classes & table of means/variances*
Asset
Index as
Mean
Std Dev
Class
Proxy
Return (%)
(%)
US Equity
Russell 3000
6.84
14.99
Dev Mkt Equity
MSCI
EAFE+Canada
6.65
16.76
Emerging Mkt
Equity
MSCI EM
7.88
23.30
Private Equity
Wilshire LBO
12.76
44.39
Hedge Funds
HFR Mkt Neutral
5.28
10.16
• Used 5 asset model due to
–computational complexity
–optimal portfolio with non-trivial
weights in each asset class
*Correlation matrix displayed in our paper
• Assumed normal returns
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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
• The optimal weights of the 5 asset classes for quadratic utility were:
19.4% US Equity, 22.2% Developed Mkt, 18.5% Emerging Mkt, 15.6%
Private Equity, 24.3% Hedge Funds
• Ran 10,000 iteration Monte Carlo simulation over 10 year period for all
three utility functions [result of quadratic utility shown below]
Quadratic Utility
Ideal
Optimal DP
5% Tolerance
Annual
Quarterly
Monthly
No Trading
Trading
Cost
Suboptimality
Cost
Aggregate
Cost
Net
Returns
Standard
Deviation
Utility
Shortfall
(bps)
(bps)
(bps)
(%)
(%)
(utils x 104)
0.00
4.04
7.39
6.84
13.68
23.66
0.00
0.00
1.72
0.70
1.55
0.28
0.00
71.72
0.00
5.75
8.09
8.39
13.96
23.66
71.72
7.45
7.40
7.37
7.40
7.32
7.22
6.77
14.84
14.86
14.83
14.94
14.85
14.84
14.96
0.00
5.55
8.03
8.24
14.28
23.72
71.36
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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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Sensitivity – US Equity Returns
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Sensitivity – Correlation
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Sensitivity – US Equity Standard Deviation
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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 > 0
Impact of short sales
Mean-reverting returns
QQQQ
PFGAX
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Conclusions
• Portfolio rebalancing theory is quite basic…rebalance
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
Acknowledgements:
Sebastien Page, State Street
Mark Kritzman, Windham Capital Management
for helpful and insightful comments (work initiating
from a project for a course in the MIT Sloan School
taught by Mark Kritzman).
Questions?