To Rebalance or Not to Rebalance: Portfolio risk may be larger than you think!∗

We show that significant portfolio return and variance biases arise when adopting a rebalancing strategy rather than using a buy-and-hold approach in dealing with portfolios spanning across multiple periods. We extend the result in Liu and Strong [2008] for bias in average portfolio returns, and derive bias in variance of portfolios. We show that the magnitude of portfolio variance bias, defined as a difference between the variance of portfolio constructed using rebalanced returns and the decomposed buy-and-hold returns, depends on average portfolio returns, the average returns of its constituents, as well as the autocovariances of the portfolio and its individual stocks. Empirical evidence based on S&P500 constituents for the period from 2003 to 2011 confirms that bias in variance of portfolios can become significant. In particular, we observe negative and significant bias during 2003, 2005 and 2010, and positive and significant bias in more turbulent 2008 and 2011. The existence of portfolio variance biases have important implications not only in evaluating the risk of such portfolios, but also in measuring their performance (e.g., when using Sharpe ratio). We vary the frequency of price quotations and estimate average return and variance biases for 5-minute, daily, weekly and monthly data. Our findings indicate that one should exercise caution when adopting a rebalancing strategy when dealing with portfolio returns, as resulting biases can lead to spurious results when analyzing investment strategies or testing asset pricing models. A popular methodology adopted by researches when dealing with portfolio returns is a simple rebalancing strategy, which suggests to rebalance portfolio back to its initial weights at the beginning of each period. Despite its simplicity and popularity, this strategy may lead to mis-estimation of decomposed returns and, as such, cumulating ∗We are grateful to AFAANZ for funding the project. †Corresponding author: Email: Vitali.Alexeev@utas.edu. au; Phone: +61 3 6226 2335, Fax: +61 3 6226 7587. these returns may not reflect the wealth effects to a buy-and-hold investor. There are several unfavorable implications for investors who adopt such strategy when estimating their multi-period portfolio returns, including • frequent rebalancing back to original weights is impractical due to prohibitive transaction costs; • the holding period return of the portfolio is not captured appropriately; • decomposed multi-period returns do not adequately represent the risk measured by the variance, resulting in overand underestimation of risk, which in turn leads to • misleading inference about the risk-adjusted performance of the portfolio (e.g., when measured by Sharpe ratio). Our contribution to the existing literature is methodological and empirical. Defining bias as a difference between estimates of portfolio constructed using rebalanced returns and the decomposed buy-and-hold returns, we derive the bias in portfolio variances, extending the result presented in Liu and Strong [2008] for the bias in portfolio mean returns. We first present the results for the two-period example, and then generalize it to a multi-period case. We show that, similarly to the bias in portfolio mean returns, bias in portfolio variance can take significant (either positive or negative) values, depending on the time-series properties of portfolio returns and the returns of its constituents. We apply the proposed methodology to portfolio returns computed using constituents of the S&P 500 index. Stocks in our constructed portfolios are selected randomly without replacements with the number of stocks in portfolios varying from 1 to 80. We consider a time frame from 2 January 2003 to 30 December 2011 to show how periods of increased volatility, observed in financial markets during the global financial crisis (GFC), impact the estimated portfolio variance resulting in large biases. We find that variance bias converges to a stable figure as the number of assets in portfolios increases, signifying systematic nature of the bias. In particular, in a welldiversified portfolio of 50 assets we observe significant negative biases during 2003, 2005 and 2010 and significantly positive bias in more turbulent 2008 and 2011. The existence of portfolio variance biases in these time periods have important implications not only when evaluating the risk of such portfolios, but also when assessing their performance by means of the coefficient of variation, the Sharpe ratio or the signal-tonoise ratio. Our results indicate that one should exercise caution when assuming multiperiod rebalanced portfolio returns, as resulting biases can lead to spurious results when analyzing investment strategies or testing asset pricing models.