Biases in variance of decomposed portfolio returns

• Published date: 01 December 2021
• Author: Vitali Alexeev,Katja Ignatieva
• Date: 01 December 2021
• DOI: http://doi.org/10.1111/irfi.12319

ORIGINAL ARTICLE

Biases in variance of decomposed portfolio

returns

Vitali Alexeev

1

| Katja Ignatieva

2

1

Finance Discipline Group, UTS Business

School, University of Technology Sydney,

Sydney, New South Wales, Australia

2

School of Risk & Actuarial Studies, University

of New South Wales, Sydney, New South

Wales, Australia

Correspondence

Vitali Alexeev, Finance Discipline Group, UTS

Business School, University of Technology

Sydney, Sydney, NSW 2007, Australia.

Email: vitali.alexeev@uts.edu.au

Funding information

Accounting and Finance Association of

Australia and New Zealand, Grant/Award

Number: 2014/2015

Abstract

Significant portfolio variance biases arise when contrasting

multiperiod portfolio returns based on the assumption of

fixed continuously rebalanced portfolio weights as opposed

to buy-and-hold weights. Empirical evidence obtained using

S&P 500 constituents from 2003 to 2011 demonstrates

that, compared with a buy-and-hold assumption, applying

fixed weights led to decreased estimates of portfolio volatil-

ities during 2003, 2005 and 2010, but caused a significant

increase in volatility estimates in the more turbulent 2008

and 2011. This discrepancy distorts assessments of portfo-

lio risk-adjusted performance when inappropriate weight

assumptions are employed. Consequently, these variance

biases have effect on statistical inference in factor models

and may result in erroneous portfolio size recommendations

for adequate diversification.

KEYWORDS

buy-and-hold strategy, high-frequency data, portfolio

diversification, portfolio risk

JEL CLASSIFICATION

G11; C58; C63

1|INTRODUCTION

A common approach in the finance literature for calculating multiperiod portfolio returns is to adopt a rebalancing

strategy that maintains a fixed weight of each asset in a portfolio at any time.

1

In contrast, if a buy-and-hold strategy

is assumed, asset weights may result in allocations far from the initial distribution when price fluctuations of some

Received: 10 September 2018 Revised: 22 April 2020 Accepted: 22 May 2020

DOI: 10.1111/irfi.12319

© 2020 International Review of Finance Ltd. 2020

1152 International Review of Finance. 2021;21:1152–1178.wileyonlinelibrary.com/journal/irfi

portfolio constituents outperform others. This is especially pertinent for longer investment horizons. To illustrate

this, Figure 1 presents portfolio weight dynamics for the fixed weight (continuous rebalancing) and the buy-and-hold

strategies. For a selection of stocks in the top panels, both strategies maintain similar allocations over time and any

differences in portfolio mean returns and portfolio variances are expected to be negligible. On the contrary, the bot-

tom panels show that for a different selection of stocks in the portfolio, the buy-and-hold strategy may lead to a

portfolio that is not well diversified (right bottom panel). In fact, one could argue that this portfolio behaves similar

to a two-stock portfolio toward the end of the period. In this case, large biases in both the average portfolio return

and the portfolio variance may be expected.

2

In evaluating portfolio performance using multiperiod portfolio returns,

an appropriate assumption on asset weights must be employed to avoid biases in estimates of the first and second

moments of portfolio returns. Estimates of portfolio average return and risk will depend on whether the assumption

of fixed or buy-and-hold weights is employed. Buy-and-hold weights ensure that compounding the decomposed

multiperiod portfolio returns yields the returns earned by an investor who holds the portfolio. In contrast, studies

that employ fixed portfolio weights for simplicity, often inadvertently assume a rebalancing frequency matching that

2004 2005 2006 2007 2008 2009 2010 2011

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Stock weights

Time

Portfolio weight breakdown

(a)

(b)

2004 2005 2006 2007 2008 2009 2010 2011

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Stock weights

Time

Portfolio weight breakdown

2004 2005 2006 2007 2008 2009 2010 2011

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0.1

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0.9

Stock weights

Time

Portfolio weight breakdown

2004 2005 2006 2007 2008 2009 2010 2011

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0.1

0.2

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Stock weights

Time

Portfolio weight breakdown

FIGURE 1 Portfolio weight dynamics for rebalanced (left panels) and buy-and-hold (right panels) strategies.

Stocks are allocated equal proportions at inception. The rebalancing is performed daily while the buy-and-hold

portfolios are held for the entire period. (a) Small bias example. Portfolio comprises AEP.N, AIG.N, AIV.N, AMGN.

OQ, APA.N, APC.N, APH.N, ASH.N (company names associated with the listed Reuters Identification Codes can be

found in the Supporting Information Appendix). Provided they had been active for the full period from January 2003

to December 2011, stocks were chosen in alphabetical order. Biases in portfolio mean return and portfolio variance

are expected to be negligible since both strategies maintain similar portfolio composition throughout the period.

(b) Large bias example. AKAM.OQ and ATI.N are added to the list of eight stocks in the panel above. The buy-and-

hold portfolio (on the right) is not as well-diversified as the rebalanced portfolio (on the left). Large biases in portfolio

mean return and portfolio variance are expected

ALEXEEV AND IGNATIEVA 1153