Portfolio Rebalancing

Edward E. Qian

• Publisher: Chapman and Hall/CRC
• Year: 2018
• eBook ISBN: 9781351647007
• Edition: 1
• Topic: Mathematics
• Subtopic: Finance
• Index: Mathematics

The goal of Portfolio Rebalancing is to provide mathematical and empirical analysis of the effects of portfolio rebalancing on portfolio returns and risks. The mathematical analysis answers the question of when and why fixed-weight portfolios might outperform buy-and-hold portfolios based on volatilities and returns. The empirical analysis, aided by mathematical insights, will examine the effects of portfolio rebalancing in capital markets for asset allocation portfolios and portfolios of stocks, bonds, and commodities.

CHAPTER 1

Introduction

Portfolio rebalancing is an essential part of portfolio management. Although portfolio rebalancing can be applied to any portfolios, such as asset allocation, equity, fixed income, commodity, and hedge fund portfolios, its earliest application was in strategic asset allocation portfolios. These multi-asset portfolios are chosen by investors to meet their long-term return objective and risk tolerance after taking into account expected returns and risks of underlying asset classes. Investors are expected to adhere to the strategic asset allocation, which also serves as a performance benchmark over the appropriate investment horizon.

1.1 Risk Management

The main objective of portfolio rebalancing, whether it is done on a calendar basis or with a threshold, is risk management. Specifically, the aim of portfolio rebalancing is to maintain portfolio risk within risk tolerance. Without portfolio rebalancing, differences in asset returns can cause an asset allocation portfolio to drift toward a portfolio that has a different risk profile in terms of total portfolio risk and risk contributions. Take, for example, a 60/40 portfolio with 60% of capital invested in stocks and 40% of capital invested in bonds. Stocks have higher risks than bonds and thus tend to have higher expected and often realized returns than bonds. As a result, a passive approach, which is often referred to as a buy-and-hold approach, would lead the weight in stocks to drift higher and the weight in bonds to drift lower. As a simple numerical illustration, suppose stocks and bonds have annualized returns of 10% and 5%, respectively, during a five-year period. Left alone for five years, a 60/40 portfolio at the beginning of the period will be close to a 65/35 portfolio. Because of higher allocation to stocks—a higher-risk asset—the 65/35 portfolio has higher risk than the original 60/40 portfolio. Furthermore, a 5% overweight in stocks at the expense of bonds exposes the portfolio to additional risks in stocks. Of course, the bias toward stocks caused by portfolio drift can be beneficial to portfolio returns, since more often than not, stocks tend to outperform bonds. In a normal market environment, higher risks usually lead to higher returns. In fact, many asset managers intentionally embed this bias into their investment process and then dubiously claim the excess return as alpha, which is actually reward for additional risk.

Of course, this additional risk in stocks can also be detrimental at times. When a bear market or market crash occurs, the 5% overweight in stocks can cause additional drawdown for asset allocation portfolios. Imagine that at the beginning of 2008, we have two portfolios: one is the 60/40 portfolio that had been rebalanced, and the other is the 65/35 portfolio. During the year 2008, stocks lost about 40% of their value as measured by the Morgan Stanley Capital International (MSCI) world index, while bonds gained about 10% as measured by the World Government Bond Index (WGBI). For the year 2008, the 60/40 portfolio was down 20%, and the 65/35 portfolio was down 22.7% with an additional loss of 2.7%. Indeed, even though in this case a buy-and-hold approach might lead to higher return over the long term relative to a rebalanced portfolio by taking on more risks, it might suffer negative excess returns in any given period. A systematic portfolio rebalancing approach can help mitigate those losses.

1.2 Rebalancing Alpha

Does portfolio rebalancing necessarily lead to a return advantage compared with a buy-and-hold approach, as we indicated in the last section? Can portfolio rebalancing generate a rebalancing alpha? In other words, can a rebalanced portfolio with lower risk outperform a buy-and-hold portfolio with higher risk? To some investors, this rebalancing alpha is another objective of portfolio rebalancing. There have been cases in which some investments have been able to generate higher returns with lower risks versus standard indices. Examples include value stock and low-volatility stocks versus capitalization-weighted indices.

The previous example hinted at this possibility. At the end of 2008, the 60/40 portfolio became a 45/55 portfolio with 45% in stocks and 55% in bonds because of dramatic underperformance of stocks. Meanwhile, the 65/35 portfolio became a 51/49 portfolio. What would happen if we rebalanced the first portfolio to 60/40 again and did not rebalance the second portfolio? It turns out that stocks rebounded strongly in 2009, with the MSCI world index returning about 23% and the WGBI index returning about 1%. The 60/40 portfolio saw a return of 14.2%, while the 51/49 portfolio saw a return of 12.2%. Hence, the rebalanced portfolio outperformed the buy-and-hold portfolio again in 2009 by 2%. However, the fact that the rebalanced 60/40 portfolio performed better than the buy-and-hold portfolio during the years 2008 and 2009 is a special case. First, the capital market was in turmoil during and after the global financial crisis. Second, the investment horizon of two years is simply too short to prove the existence of rebalancing alpha.

To answer the question of whether portfolio rebalancing adds value or not, we need more research in terms of both empirical study and theoretical investigation. Unfortunately, no empirical study of portfolio rebalancing has produced any concrete answer to the question. In fact, many empirical studies have added more confusion to the topic, because some studies show that portfolio rebalancing added value, while other studies show just the opposite. This is not unexpected, since any empirical study is specific to asset classes, portfolio weights, and characteristics of asset returns during the time periods chosen. General claims about rebalancing alpha or the lack of it based on one specific empirical study cannot be substantiated. The results of empirical studies raise a further question, however: why portfolio rebalancing adds value in some cases but not in other cases. Perhaps these questions can be tackled by theoretical research on portfolio rebalancing.

1.3 Diversification Return, Volatility Effect

One of the important and unfortunately often misunderstood theoretical concepts in portfolio rebalancing is diversification return. Diversification return refers to the difference between the geometric return of a fixed-weight portfolio (i.e. a rebalanced portfolio) and the weighted average of geometric returns of the underlying investments. Take the 60/40 portfolio as an example. The diversification return equals the geometric return of a rebalanced 60/40 portfolio minus the weighted sum of the geometric returns of stocks and bonds, with the weights being 60% for stocks and 40% for bonds.

Mathematically, it can be proved that for a long-only portfolio, diversification return is always non-negative. The reason why it is called diversification return has to do with portfolio diversification of risk, since the mathematical proof hinges on the fact that the variance of a fixed-weight portfolio is always less than or equal to the weighted average of variances of underlying investments. The term diversification is not the source of misunderstanding.

The misunderstanding of diversification return is probably caused by the term return. Some investors and researchers falsely think that diversification return is the return difference between the geometric return of the rebalanced portfolio and the geometric return of the corresponding buy-and-hold portfolio. Since diversification return is always non-negative for a long-only portfolio, many are led to believe that diversification return is rebalancing alpha. In other words, portfolio rebalancing always adds value for a long-only portfolio.

This notion is wrong, because the weighted average of geometric returns of underlying investments is not the geometric return of the buy-and-hold portfolio. In fact, it is generally not the return of any portfolio. In some sense, it is a mathematical construct based on underlying returns, to which the fixed-weight portfolio is compared. It turns out that this comparison is analytically tractable through the variance of the fixed-weight portfolio and the variances of underlying investments. Nevertheless, diversification return is not rebalancing alpha, because it is not the return difference between the rebalanced portfolio and the buy-and-hold portfolio.

It is in fact half of rebalancing alpha. To complete the analysis, the next logical step is to compare the geometric return of the buy-and-hold portfolio with the same weighted average of underlying returns. It can be shown that for a long-only portfolio, the geometric return of a buy-and-hold portfolio is also always greater than or equal to the weighted average of the geometric returns of underlying components. If we denote the geometric return of the fixed-weight portfolio by $g_{\text{FW}}$, that of the buy-and-hold portfolio by $g_{\text{BH}}$, and the weighted average of geometric returns of underlying investments by $\overline{g}$, we have $g_{\text{BH}}\ge \overline{g}$ and $g_{\text{FW}}\ge \overline{g}$. The triangulation of three terms can only be completed by comparing the two differences: $g_{\text{BH}}-\overline{g}$ versus $g_{\text{FW}}-\overline{g}$. We call them the return effect and the volatility effect of portfolio rebalancing, respectively. If the return effect is less than the volatility effect, rebalancing alpha is positive. But if the return effect is greater than the volatility effect, rebalancing alpha would be negative. The establishment of this framework allows a thorough analysis of both effects to determine the overall rebalancing alpha.

1.4 Serial Correlation and Rebalancing Alpha

It is natural to suspect that rebalancing alpha is related to serial correlations of asset returns. Specifically, positive serial correlations or return momentum lead to negative rebalancing alpha, while negative serial correlations or return reversal lead to positive rebalancing alpha. This is supported by our example of the 60/40 portfolio during the year 2008 and 2009. In 2008, stocks underperformed bonds significantly, and the pattern reversed in 2009, when stocks outperformed bond significantly. There was a strong reversal of relative return between stocks and bonds. The rebalance at the end of 2008 of buying stocks (previous loser) and selling bonds (previous winner) helped the portfolio in 2009, when winner and loser reversed. If, however, stocks had underperformed bonds again in 2009, or if there had been a trend in the relative return between the two assets, the rebalance at the end of 2008 would have had a negative impact on the portfolio.

So, does positive rebalancing alpha necessitate mean reversion in asset returns? This question is frequently debated among practitioners and academics. The answer seems to be yes, at least from an academic perspective. It can be proved statistically that if underlying returns are serially uncorrelated and have the same expected value, then the expected returns of a fixed-weight portfolio and a buy-and-hold portfolio would be identical; that is, no rebalancing alpha.

This is a very clear and strong result in theory. In practice, however, no returns fit these statistical assumptions perfectly. What can be expected of serial correlations of returns when rebalancing alpha is either positive or negative? The analysis of volatility and return effects with approximations seems inadequate to answer this question. However, an exact treatment of rebalancing alpha can be used for studying this question. We shall provide evidence that a positive rebalancing alpha does require some degree of mean reversion.

The structure of serial correlations of different lags can also have an impact on rebalancing alpha of varying frequency and/or thresholds. For example, if asset returns exhibit short-term momentum and long-term reversal, it is probably optimal to carry out portfolio rebalancing with a longer horizon or with a larger threshold. In this area of research, one often has to use numerical simulations to analyze different options.

1.5 New Topics in Portfolio Rebalancing

Most previous research on portfolio rebalancing focuses on long-only portfolios. This is quite understandable, because most investors in the past invested in traditional long-only portfolios without leverage. In recent years, there has been growing dev...

Table of contents

• Cover
• Half-Title
• Series
• Title
• Copyright
• Dedication
• Contents
• Preface
• Chapter 1 ■ Introduction
• Chapter 2 ■ A Brief Review of Portfolio Theory
• Chapter 3 ■ Portfolio Rebalancing
• Chapter 4 ■ Volatility Effect and Return Effect
• Chapter 5 ■ Analysis of Volatility Effect
• Chapter 6 ■ Analysis of Return Effect
• Chapter 7 ■ Analysis of Rebalancing Alpha
• Chapter 8 ■ Asset Allocation Portfolios
• Chapter 9 ■ Asset Class Portfolios
• Chapter 10 ■ Rebalancing Alpha and Mean Reversion
• Chapter 11 ■ Risk and Return of Rebalancing Effects
• Chapter 12 ■ Threshold Rebalancing
• Bibliography
• Index