Optimal Rebalancing Strategy Optimal Rebalancing Strategy for Pension Plansfor Pension Plans

A Presentation to State Street Associates15.451 Financial Engineering Proseminar

MIT Sloan School of Management

November 18, 2004

Marius Albota Josh GroverTom Schouwenaars

Walter SunLi-Wei ChenAyres Fan

Ed Freyfogle

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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.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

• 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 moneyGetting this right would be worth lots (and we mean lots) of money

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Presentation Outline

• Simple Example

• Our Solution

• Methodology

• Two Asset Model

• Multi-Asset Model

• Sensitivity Analysis

• Conclusion

• Future Research

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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!!!

0.0%

2.0%

4.0%

6.0%

8.0%10.0%

12.0%

14.0%

16.0%

18.0%

8/17

8/31

9/14

9/28

10/12

10/26 11

/9

QQQ S&P500

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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

-2.0%0.0%2.0%4.0%6.0%8.0%

10.0%12.0%14.0%16.0%18.0%

8/17

8/31

9/14

9/28

10/12

10/26 11

/9

PFGAX QQQ

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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.

When and how to rebalance is complicated.

Transaction costs make it much more difficult.

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Our Solution

• In theory when to rebalance is easy:

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). • It is difficult to know the benefit of rebalancing.

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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 costs

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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

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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

• 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 functionsQuadratic Utility

Trading Suboptimality Aggregate Net Standard UtilityCost Cost Cost Returns Deviation Shortfall(bps) (bps) (bps) (%) (%) (utils x 10,000)

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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Simulation Results (cont.)

$2.8 MM in savings!!! $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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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

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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

Thanks:Thanks:

Sebastien Page, VP State Street Mark Kritzman, Windham Capital Management

Questions?Questions?