Benchmarking and Rebalancing* Daniel Gabay (CNRS-EHESS & ESILV) Daniel Herlemont (ESILV & yats.com) *This work is part of a project supported by a research grant from Institute Europlace of Finance which is gratefully acknowledged 29 juin 2007

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Benchmarking and Rebalancing Benchmark: strategic allocation
Passive mgt: Buy & Hold benchmark







With time, drift in allocation:  




Index funds Over exposition to over performing assets Risk drift

Rebalancing: active mgt strategy 

maintain benchmark weights

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Rebalancing vs Buy&Hold


Rebalancing: common practice to reset (monthly) against risk drift, but  

Transactions costs Missed trends: sell the winners (contrarian)




Main message: rebalancing improves portfolio performance (in probability)




J.-F. Boulier & C. Leclerc (AFFI 2004) « vertus et performance du rebalancement » Empirical studies J.Portfolio Mgt (2002)

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Benchmarking and Rebalancing








CRP: Constantly Rebalanced Portfolio « fixed mix » π = (π1,π2, …,πn) Idea goes back to Kelly (1956)  « bet each time a constant fraction of wealth » Particular benchmarks considered: considered:  





Market portfolio: πi relative capitalization of asset i Value line: equally weighted portfolio πi = 1/n

Theme: Theme: compare CRP and BH for such strategies π Next, Next, for some non constant weight strategies

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Outline


Rebalancing vs B&H: a simple example   





Pumping 2 assets Extension to n assets 





Multi asset Rebalancing vs B&H

CRP vs B&H for different benchmarks Extension: nonconstant mix strategies  




Volatility pumping Portfolio Excess Growth Constantly Rebalanced Portfolio formulae

Entropic portfolio Diversity weighted portfolio

Conclusions

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Rebalancing vs B&H: a simple simulated example


Invest 






1/2 in cash riskfree rate = 0% 1/2 in BS risky asset drift = 12,5% volatility = 50% S(0) = 1

Equally weighted portfolio

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Example analysis: Volatility pumping











CRP underperforms BH only in early periods CRP captures « excess growth rate » g ≈ 3% Increasing benefit with time CRP deteriorates when S(t) follows a trend, i.e. diverges from 1

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Asset growth rate


Lognormal diffusion model for asset price dS(t)/S(t) dS(t)/S(t) = b dt + σ dB(t) S(t) = exp( exp(gt + σ√t z(t)) where g = b - σ 2/2 is the asset growth rate lim 1/t (log S(t) - gt)) = 0 as t → +∞ (logS Remark: Remark: in example g = 0

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Rebalanced Portfolio Excess Growth Portfolio fix mix: mix: π in asset, asset,1- π in cash d(log WCRP(t)) = (πb - π2 σ2/2) dt + πσ dB(t) d(logW = g*dt + π d(log S(t)) d(logS g* = π(1(1-π)σ 2/2 is the excess growth rate  Remarks: Remarks: 1) Important: g* increases with volatility σ Luenberger’ Luenberger’s Volatility pumping (1998) Fernholz’s Stochastic portfolios (2002) 2) g* is maximum for π =1/2 

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CRP formula (A.J. Wise 1993) d(log WCRP(t)) = g*dt + π d(log S(t)) d(logW d(logS BH W (t) = 11- π + πS(t)

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Rebalancing vs B&H revisited


in case π =1/2, g = 0






For small t, probability to overperform > 68,3%

For small t, independently of π

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Rebalancing vs B&H: empirical tests




Nasdaq countercounter-example: example: short term trends deteriorate volatility pumping! pumping!

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Rebalancing vs B&H: long term


For large t (always in case π =1/2, g = 0) where  




Probability overover-performance increases with time, tends to 1 Underperformance less frequent, Under frequent, but can be severe!

PeroldPerold-Sharpe (1988): contrarian strategies leads to concave paypay-off: no downside protection; protection; do poorly in up markets

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Pumping 2 assets: simulation






Equal weights Asset 1: drift = 8% vol = 30% Asset 2: drift = 12% vol = 50% Correlation = - 0.3

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Pumping 2 assets: simulation

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Pumping 2 assets: analysis






Same as before: before: equivalent to pumping cash and derivative S1/ S2 Benefit of rebalancing increases as spread decreases Same formula

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Extension to n assets


CRP excess growth rate:




Introduce Fernholz’s generating function




CRP formula 




Wise (1993)

Remark: Remark: W(t) depends only on S(t), not on path

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Multi asset Rebalancing vs B&H


Compare Rebalancing vs B&H




First term: term: excess growth accumulated with time Second term: term: always Wµ(T)) = 1 for a sufficiently large horizon T (asymptotic arbitrage)

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Conclusions


Constant rebalancing to benchmark allocation often improves performance, especially in the long term; but poor results in strong market trends




More sophisticated active management with functionally generated portfolio exploiting market diversity is promising 

Fernholz’s experience at INTECH

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