2. Optimal Rebalancing Strategy for Institutional Portfolios 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

3. 2 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.

4. 3 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.

5. 4 Presentation Outline Simple Example Our Solution Two Asset Model Multi-Asset Model Sensitivity Analysis Conclusion Future Research

6. 5 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.

7. 6 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.

8. 7 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.

9. 8 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

10. 9 Utility Functions Quantify risk preference Assume three possible utilities

11. 10 Certainty Equivalents Given a risky portfolio of assets, there exists a risk-free return r CE (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 x opt is equivalent to 50 bps per month –A sub-optimal portfolio x sub is equivalent to 48 bps per month –On this portfolio, that difference amounts to $2 million per month

12. 11 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 r1r1 Accept if r 1 >E(J 2 (r 2 )) Roll J 1 (r 1 ) = max( r 1, E(J 2 (r 2 )) ) r2r2 Roll Accept if r 2 >3.5 r3r3 J 2 (r 2 ) = max( r 2, E(J 3 (r 3 )) ) = max( r 2, 3.5 ) J 2 (r 2 ) – expected benefit at time 2, given roll of r 2

13. 12 Dynamic Programming Examine costs rather than benefit J t (w t ) is the “cost-to-go” at time t given portfolio w t Trade to w t+1 (optimal policy) –When w t+1 = w t, no trading occurs Current period tracking error Cost of TradingExpected future tracking error

14. 13 Data and Assumptions Given annual returns for 5 asset classes & table of means/variances* Assumed normal returns AssetIndex asMeanStd Dev ClassProxyReturn (%)(%) US EquityRussell 30006.8414.99 Dev Mkt EquityMSCI EAFE+Canada 6.6516.76 Emerging Mkt Equity MSCI EM7.8823.30 Private EquityWilshire LBO12.7644.39 Hedge FundsHFR Mkt Neutral5.2810.16 Used 5 asset model due to –computational complexity –optimal portfolio with non-trivial weights in each asset class *Correlation matrix displayed in our paper

15. 14 Optimal Portfolios Calculated efficient frontier from means and covariances Performed mean-variance optimization to find the optimal portfolio on efficient frontier for each utility

16. 15 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

17. 16 Two Asset Model

18. 17 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

19. 18 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!!!

20. 19 Sensitivity – US Equity Returns

21. 20 Sensitivity – Correlation

22. 21 Sensitivity – US Equity Standard Deviation

23. 22 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

24. 23 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

25. 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).

26. Questions?