How much should portfolios shrink?

• DOI: http://doi.org/10.1111/fima.12282
• Author: Chulwoo Han
• Published date: 01 September 2020
• Date: 01 September 2020

DOI: 10.1111/fima.12282

ORIGINAL ARTICLE

How much should portfolios shrink?

Chulwoo Han

Department of Finance and Economics, Durham

Business School, Durham, UK

Correspondence

ChulwooHan, Department of Finance and Eco-

nomics,Durham Business School, Mill Hill Lane,

DurhamDH1 3LB, UK.

Email:chulwoo.han@durham.ac.uk

Abstract

This paper develops a portfolio model that penalizes the devia-

tion from a reference portfolio. The proposed model renders a

robust portfolio that performs superior under parameter uncer-

tainty. Penalizing the deviation also improves the performance of

existingshrinkage portfolio models that are suboptimal due to model

parameter uncertainty.The equal-weight portfolio turns out to be a

better reference portfolio than the currently holding portfolio even

in the presence of transactioncosts. A data-driven method for deter-

mining the degree of penalization is offered. Comprehensive simula-

tion and empirical studies suggest that the proposed model signifi-

cantly outperforms various existing models.

1INTRODUCTION

Optimal portfolio choice under parameter uncertainty and transaction costs is a centralproblem in the portfolio liter-

ature, and there has been a significant amount of effort dedicated to this problem.

One pillar has been formed by the Bayesian approach: for example, Klein and Bawa (1976); Brown (1976, 1978);

Jorion (1986); Black and Litterman (1992); Pástor(2000); Pástor and Stambaugh (2000), among others. For a review of

Bayesianmodels, the reader is referred to Avramov and Zhou (2010). More recently, the robust optimization that opti-

mizes portfolio under a worst-case scenario has become popular: for example, Goldfarb and Iyengar (2003); Fabozzi,

Kolm, Pachamanova, and Focardi (2007); Cao, Han, Hirshleifer, and Zhang (2009); Ceria and Stubbs (2006). Kan and

Zhou (2007) and Tuand Zhou (2011) optimally combine two or more portfolios to minimize the expected utility loss.

Incorporating transaction costs has also been found to reduce the sensitivity and improve the performance after

transaction costs: for example, Gârleanu and Pedersen (2013); DeMiguel, Martín-Utrera, and Nogales (2015). Other

approaches impose weight constraints (Jagannathan & Ma, 2003) or use a shrinkagemethod for parameter estimation

(Ledoit & Wolf,2004).

Although these models alleviate the problems arising from parameter uncertainty and demonstrate superior per-

formance to the classical mean-variance model, DeMiguel, Garlappi, and Uppal (2009) find that none of the portfolio

models considered in their paper consistently outperforms the naïve, equal-weight portfolio. Their work has triggered

many studies that challenge the equal-weight portfolio: for example, Tuand Zhou (2011); Kirby and Ostdiek (2012);

Bessler,Opfer, and Wolff (2017). Their evaluation method comparing risky-asset-only portfolios derived from optimal

portfolios has also been criticized as being unfair to some models (see, e.g., Kirby and Ostdiek (2012) and Kan, Wang,

c

2019 Financial Management Association International

Financial Management. 2020;49:707–740. wileyonlinelibrary.com/journal/fima 707

708 HAN

and Zhou (2016)). Still, most optimal strategies seem to struggle to outperform the naïve strategyconsistently across

assets and time.

This paper addresses parameter uncertainty by developing a portfolio model that penalizes the deviation from a

reference portfolio (deviation penalty, henceforth), where the reference portfolio can be anyportfolio known at the

time of rebalancing, such as the current portfolio or the equal-weight portfolio. By penalizing the deviation from a

reference portfolio, this model reduces the time series variation of portfolio weights and generates a portfolio that is

robust to estimation errors. Although the model can be considered a shrinkage model, existing shrinkage models can

also benefit from the deviation penalty as illustrated below.

This paper offers a data-driven method to determine the degree of penalization and finds that the proposed model

significantly outperforms many existing models in terms of the certainty equivalent (CE) and Sharpe ratio before and

after transaction costs. The optimal degree of penalization is strikingly high when compared to the shrinkage levels of

other shrinkage models, especially when the input parameters are subject to large estimation errors.

Comprehensivesimulation and empirical studies involving 13 datasets and a sample period of over 60 years suggest

that the proposed model is superior when compared to various existing models, such as the equal-weight portfolio,

the market portfolio, and other shrinkage models. Robustness tests indicate that the proposed model continues to

outperform other models in different circumstances: for example, during different sample periods, during recessions,

and when using different sample sizes for input parameter estimation.

Apart from introducing a new portfolio model that exhibits superior performance, this paper makesseveral impor-

tant contributions to the extant literature.

Oneimportant contribution of the paper is to demonstrate, both theoretically and empirically, that the equal-weight

portfolio is a more effective reference portfolio than the current portfolio. Although penalizing the deviation from the

currentportfolio is helpful to some extent, especially when trades are subject to transaction costs, its effect on portfolio

performanceis found to be rather trivial when compared to the effect of penalizing the deviation from the equal-weight

portfolio. Shrinking toward the equal-weight portfolio renders a less volatile portfolio with better performance. Coun-

terintuitively,the equal-weight portfolio also incurs fewer transaction costs. Using the current portfolio is certainly a

more effectiveway of reducing turnover for a single period. However, because the current portfolio can be distant from

the true optimal portfolio under parameter uncertainty,shrinking toward it can cause higher turnover and transaction

costsin the long run. As penalizing the deviation from the current portfolio is similar to accounting for transaction costs,

this finding is contrary to the earlier findings that accounting for transaction costs in portfolio optimization enhances

portfolio robustness and performance: for example, Gârleanu and Pedersen (2013); DeMiguel et al. (2015); Olivares-

Nadal and DeMiguel (2018).

Another important contribution of the paper is to demonstratethat existing shrinkage models, for example, Kan and

Zhou(2007) and Tu and Zhou (2011), are suboptimal and can be improved substantially when augmented with the devi-

ation penalty. These models combine two or more portfolios so that the expectedout-of-sample utility is maximized.

However,the coefficients on the portfolios (model parameters) are nonlinear functions of unknown input parameters

and,as such, inherit their uncertaintyresulting in worse than expected performance even when the underlying assump-

tions are correct. That is, just likethe mean-variance optimal portfolio is not optimal when the mean and the covariance

matrix are subject to estimation errors, the shrinkage models are not optimal when the model parameters are subject

to estimation errors inherited from the mean and the covariance matrix. If any of the assumptions, such as i.i.d. normal

returns are violated, which is very likely,the problem is exacerbated. Although the impact of model parameter uncer-

tainty can be substantial, the extant literature has failed to recognize this. This paper finds that the deviation penalty

increases the robustness of existing shrinkage portfolios and alleviates the performance deterioration due to model

parameter uncertainty.

The rest of the paper is organized as follows. Section 2 develops portfolio models with the deviation penalty.A cal-

ibration method to determine the degree of penalization is also offered here. Section 3 describes the datasets and

portfolio models used in the empirical study. Section 4 evaluates the proposed models via simulations. Two refer-

ence portfolios, the equal-weight and the current portfolios, are examined. Section 5 carries out empirical studies that

HAN 709

compare the proposed models against various existing models, while Section 6 provides the conclusions. The imple-

mentation details of the models used in the empirical analysis and the full empirical results are provided in the accom-

panying internet appendix (Internet Appendix).

2OPTIMAL PORTFOLIO WITH DEVIATION PENALTY

2.1 Utility maximization

The quadratic utility maximization problem with the deviation penalty is givenby

max

wU(w)=w′𝜇−𝛾

2w′Σw−𝛿

2(w−w0)′G(w−w0),(1)

where 𝜇∈ℝNand Σ∈ℝN×Nare the mean and covariance matrix of Nasset returns in excess of the risk-free rate,

w∈ℝNis the portfolio weights, and 𝛾is the risk aversion coefficient of the investor.1The last term on the right-hand

side penalizes the deviation from a reference portfolio w0,where𝛿is a constant and G∈ℝN×Nis a penalty matrix. The

reference portfolio w0can be any portfolio known at the time of portfolio rebalancing: the equal-weight portfolio, wew,

and the current portfolio, wt−, are considered in this paper.

The optimal portfolio w∗that maximizes the utility function is given by

w∗=(𝛾Σ+𝛿G)−1(𝜇+𝛿Gw0).(2)

If an asset return has a large variance, its mean estimate may well have a large estimation error and it is justifiable to

penalize the weight change of such assets more severely. From this perspective,a natural choice of Gwould be the

covariance matrix, Σ. When G≡Σ, the optimal portfolio becomes a convexcombination of the Markowitz (1952) opti-

mal portfolio, wml =1

𝛾Σ−1𝜇, and the reference portfolio, w0:

w∗=𝛾

𝛾+𝛿wml +𝛿

𝛾+𝛿w0.(3)

Inorder to implement w∗, unknown 𝜇and Σmust be estimated. If the asset returns are i.i.d.normal random variables,

the maximum likelihood (ML) estimates of 𝜇and Σ,̂𝜇 and ̂

Σ, are independent of each other and have the following

distributions:

̂𝜇 ∼𝜇,Σ

T,̂

Σ∼N(T−1,Σ)1

T,(4)

where Tis the estimation window size, and and N, respectively,denote a normal distribution and N-dimensional

Wishart distribution. Toallow the case when asset returns are not i.i.d. or ̂𝜇 and ̂

Σare estimated separately,for example,

using different estimation windows, a slightly relaxed assumption,

̂𝜇 ∼𝜇,Σ

K,̂

Σ∼N(T−1,Σ)1

T,(5)

for some constant K, is made.

An unbiased estimate of the Markowitz portfolio is then given by

̂

wml =1

𝛾

̃

Σ−1̂𝜇,̃

Σ= T

T−N−2̂

Σ.(6)

1Returnsrefer to excess returns throughout the paper unless otherwise noted.