Optimal Rebalancing Strategy Optimal Rebalancing Strategy for Institutional Portfoliosfor Institutional Portfolios

Walter Sun

Joint work with Ayres Fan, Li-Wei Chen,Tom Schouwenaars, Marius Albota,

Ed Freyfogle, Josh Grover

QWAFAFEW - Boston MeetingApril 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.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.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.

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

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 trading2. Cost of not being optimal this period3. Expected future costs of our current actions

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

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 payoutSolution• Work backwards to determine optimal policy

r1

Accept if r1>E(J2(r2))

Roll• J1(r1) = max( r1, E(J2(r2)) )r2

Roll

Accept if r2>3.5

r3• J2(r2) = max( r2, E(J3(r3)) ) = max( r2, 3.5 )

• J2(r2) – expected benefit at time 2, given roll of 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

•Trade to wt+1 (optimal policy)–When wt+1 = wt, no trading occurs

Current period tracking error

Cost of Trading Expected future tracking error

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Data and Assumptions

• Given annual returns for 5 asset classes & table of means/variances*

• Assumed normal returns

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

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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

Trading Suboptimality Aggregate Net Standard UtilityCost Cost Cost Returns Deviation Shortfall(bps) (bps) (bps) (%) (%) (utils x 104)

Ideal 0.00 0.00 0.00 7.45 14.84 0.00Optimal DP 4.04 1.72 5.75 7.40 14.86 5.555% Tolerance 7.39 0.70 8.09 7.37 14.83 8.03Annual 6.84 1.55 8.39 7.40 14.94 8.24Quarterly 13.68 0.28 13.96 7.32 14.85 14.28Monthly 23.66 0.00 23.66 7.22 14.84 23.72No Trading 0.00 71.72 71.72 6.77 14.96 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!!! 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 moneyUse dynamic programming to save money

Acknowledgements: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?Questions?