1 Optimal Rebalancing Strategy for Institutional Portfolios Walter

1 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

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.

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.

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

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.

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 (40 bps, 60 bps, or flat fee)? When and how to optimally rebalance is complicated. Transaction costs make it much more difficult.

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.

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

9 Utility Functions • Quantify risk preference • Assume three possible utilities

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 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

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 • J 2(r 2) – expected benefit at time 2, given roll of r 2 • J 2(r 2) = max( r 2, E(J 3(r 3)) ) = max( r 2, 3. 5 ) Roll • J 1(r 1) = max( r 1, E(J 2(r 2)) ) Roll r 1 r 3 r 2 Accept if r 2>3. 5 Accept if r 1>E(J 2(r 2))

12 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

13 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

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

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

16 Two Asset Model

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

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!!!

19 Sensitivity – US Equity Returns

20 Sensitivity – Correlation

21 Sensitivity – US Equity Standard Deviation

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

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

24 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).

25 Questions?