Optimal Rebalancing Strategy for Pension Plans A Presentation

Optimal Rebalancing Strategy for Pension Plans A Presentation to State Street Associates 15. 451 Financial Engineering Proseminar MIT Sloan School of Management Marius Albota Li-Wei Chen Ayres Fan Ed Freyfogle Josh Grover Tom Schouwenaars Walter Sun November 18, 2004 1

2 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.

3 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

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

5 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!!!

6 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

7 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.

8 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.

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

11 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

12 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))

13 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 costs

14 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 • Assumed normal returns

15 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 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 Two Asset Model

18 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

19 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 Simulation Results (cont. ) $2. 8 MM in savings!!!

21 Sensitivity – US Equity Returns

22 Sensitivity – Correlation

23 Sensitivity – US Equity Standard Deviation

24 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 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

26 Thanks: Sebastien Page, VP State Street Mark Kritzman, Windham Capital Management

27 Questions?