What Does Rebalancing Really Achieve?

• Published date: 01 July 2016
• Date: 01 July 2016
• DOI: http://doi.org/10.1002/ijfe.1545

WHAT DOES REBALANCING REALLY ACHIEVE?

KEITH CUTHBERTSON

,

*, SIMON HAYLEY, NICK MOTSON and DIRK NITZSCHE

Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK

ABSTRACT

There is now a substantial literature on the effects of rebalancing on portfolio performance. However, this literature contains

frequent misattribution between ‘rebalancing returns’, which are specific to the act of rebalancing, and ‘diversification returns’,

which can be earned by both rebalanced and unrebalanced strategies. Confusion on this issue can encourage investors to follow

strategies that involve insufficient diversification and excessive transactions costs. This paper identifies the misleading claims

that are made for rebalanced strategies and demonstrates in theory and by simulation that the apparent advantages of rebalanced

strategies over infinite horizons give an inaccurate impression of their performance over finite horizons. Copyright © 2016 John

Wiley & Sons, Ltd.

Received 16 November 2015; Accepted 17 November 2015

JEL CODE: G10; G11

KEY WORDS: Rebalancing; diversification returns; excess growth; volatility pumping

1. INTRODUCTION

Rebalancing is a vital part of many investment strategies. Passive strategies usually involve choosing portfolio

weights according to some predetermined rule, and a key practical element of such strategies is the frequency

and hence cost of rebalancing. Only if portfolio weights are allowed to evolve over time according to the relative

returns on the component assets (a buy and hold strategy, B&H) will there be no rebalancing. Other passive strat-

egies require at least periodic rebalancing, and in the extreme, some passive strategies assume continuous

rebalancing to keep asset weights constant (an assumption that is often required to produce tractable closed form

results). Rebalancing is so widespread that a good understanding of its effects is vital.

The well-established literature on optimal growth considers portfolio weights chosen to maximize the ex-

pected geometric growth rate of the portfolio value, consistent with maximizing the expected logarithm of

final wealth (Kelly, 1956; Luenberger, 1997; Thorp, 2010; Platen and Rendek, 2010). Rebalancing is a vital

part of these strategies. Another strand of the literature considers arbitrary fixed weight portfolios, which are

compared with a B&H strategy (e.g. Fernholz and Shay, 1982; Booth and Fama, 1992; Luenberger, 1997;

Mulvey et al., 2007; Qian, 2012; Willenbrock, 2011). A key result from this part of the literature is that a

rebalanced portfolio of independently and identically distributed (IID) assets has a higher expected portfolio

growth rate than the corresponding B&H strategy even when assets follow a random walk. ‘Volatility

pumping’is a strategy that seeks to boost this effect by deliberately choosing high volatility assets and

rebalancing each period to fixed weights.

In this paper, we focus on the choice between constant portfolio weight rebalanced strategies and the corre-

sponding B&H strategies. We examine under what circumstances and for what reasons one strategy outperforms

the other. The purpose of this paper is to correct misleading claims that are widely made about the benefits of

rebalancing.

*Correspondence to: Simon Hayley, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK. E-mail: simon.hayley.1@city.ac.uk

Copyright © 2016 John Wiley & Sons, Ltd.

International Journal of Finance & Economics

Int. J. Fin. Econ. 21: 224–240 (2016)

Published online 17 February 2016 in Wiley Online Library

(wileyonlinelibrary.com). DOI: 10.1002/ijfe.1545

It is widely noted that even when there is no predictable time structure to asset returns, rebalanced portfolios

generate ‘excess growth’(defined in this literature as expected geometric portfolio growth which is greater than

the average geometric growth of the underlying assets). We demonstrate mathematically that unrebalanced portfo-

lios also generate ‘excess growth’and that growth rates of unrebalanced and rebalanced portfolios only diverge to

the extent that drift in portfolio weights gradually leaves the unrebalanced portfolio less well diversified.

Confusion on this issue appears to have arisen in part because of the difficulty in making meaningful com-

parisons between rebalanced and unrebalanced portfolios, because even when the portfolios are initially iden-

tical, the composition of the unrebalanced portfolio tends to shift over time. We derive like-for-like

comparisons between rebalanced and unrebalanced portfolios and demonstrate analytically that in the absence

of mean reversion in relative asset prices, the greater expected growth of rebalanced strategies is entirely ex-

plained by their lower portfolio volatilities rather than –as is claimed –being due to the rebalancing trades

themselves being profitable. We also demonstrate that the apparent advantages of such rebalanced strategies

over infinite horizons are very misleading indicators of performance over the finite horizons that are likely to

be of interest to investors.

This has important implications for investors. The misleading claims that are made for rebalancing encourage

investors to increase the scale of their rebalancing trades by holding volatile assets and rebalancing frequently.

More efficient portfolios can be constructed simply by diversifying effectively. Investors might rationally prefer

rebalancing strategies where they anticipate mean reversion in relative asset prices, but this is very different from

the claims in the theoretical literature that rebalancing directly boosts returns even in the absence of any such time

structure in returns.

The rest of the paper is organized as follows. Section 2 reviews the theoretical and empirical literature

on rebalancing. Section 3 shows, using analytic and simulation approaches, that under standard assump-

tions, the expected growth rate of a portfolio of risky assets (either rebalanced or B&H) is entirely ex-

plained by diversification, with no additional ‘rebalancing return’. Section 4 examines the impact o f

rebalancing in the widely cited case of a portfolio consisting of one risky and one risk-free asset. Conclu-

sions are drawn in Section 5.

2. LITERATURE REVIEW

Rebalancing is important in a wide range of situations. It is inherent in any portfolio-weighting strategy other than

capitalization-weighting and plays a part in the debate over fundamental and alternative forms of equity indexing

(Arnott et al., 2005; Kaplan, 2008; Hsu et al., 2011) and universal portfolios (Cover, 1991). Rebalancing also has a

significant impact on the growth rates of portfolios of commodity futures (Gorton and Rouwenhorst, 2006a and

2006b; Erb and Harvey 2006).

In the theoretical literature, Cheng and Deets (1971) compare the performance of B&H and rebalancing

strategies, assuming that risky asset prices follow random walks and are IID except for their different mean

returns μ

i

. The B&H strategy and the rebalancing strategy give the same expected terminal wealth when

the mean returns of all risky assets are equal. However, if at least one pair of assets has different mean

returns μ

i

≠μ

j

, then an initially equally weighted B&H strategy always gives a higher expected terminal

wealth than the corresponding rebalancing strategy (intuitively, a B&H portfolio is likely over time to give

increasing weight to the assets with higher μ

i

). This relative superiority of B&H in terms of expected

wealth is found to be larger the greater the dispersion of the μ

i

, the longer the investment horizon and

the more frequently the rebalanced portfolio is returned to equal weights.

A specific situation that is widely analysed (e.g. Fernholz and Shay, 1982; Perold and Sharpe, 1988;

Luenberger, 1997; Gabay and Herlemont, 2007) is a portfolio that is rebalanced each period to keep a fixed pro-

portion 1 πin a risk-free asset that pays zero interest. The remainder is invested in a risky asset that follows

the lognormal diffusion process S

t

= exp(gt +σW(t)), where g=μσ

2

/2 and W(t) is a Wiener process.

Hence, if g= 0, the long-term growth rate of this risky asset approaches zero as t →∞

.

However, the terminal wealth

of the rebalanced portfolio is

WHAT DOES REBALANCING REALLY ACHIEVE?225

Copyright © 2016 John Wiley & Sons, Ltd. Int. J. Fin. Econ. 21: 224–240 (2016)

DOI: 10.1002/ijfe