A new procedure for resampled portfolio with shrinkaged covariance matrix

ABSTRACT

Dealing with estimation error is an important issue when we implement the mean–variance paradigm for portfolio construction. To tackle the problem, two approaches are proposed in literature, the portfolio resampling technique introduced by Michuad and the well-known shrinkaged covariance matrix method. There are certain evidences on the advantages of shrinkaged covariance over portfolio resampling, however, it is unclear whether a combination of the two approaches could produce a better performance compared with using shrinkaged covariance alone. In this paper, we propose a new algorithm to integrated linear or nonlinear shrinkage estimation with resampled portfolio to achieve a further improvement. Our method are demonstrated via extensive simulation and application in active portfolio management process.

1. Introduction

Markowitz mean–variance (MV) optimization is a popular and convenient framework for optimal portfolio construction and asset allocations. However, in practice, it could be an unstable procedure that may result in extreme or unrealistic portfolios ([11,20]). As the inputs of covariance matrix and expected return contains estimation error, the optimal weights are inherited under uncertainty of the estimated parameters. There are two methods to handle the issue of estimation error, the shrinkaged covariance estimation approach and the resampled portfolio approach. It is well recognized that shrinkaged covariance is an improvement for the sample covariance. However, there are some controversies on the resampled portfolio approach.

The shrinkaged covariance method resemble the bias-variance tradeoff for the choice of estimation. By introducing some bias, a large amount of the variance of the estimate can be reduced, and hence an improvement is obtained. Ledoit and Wolf [13] proposed shrinkage covariance by optimally weighting average of the sample covariance and an estimated covariance matrix by assuming a one-factor model. Ledoit and Wolf [15] introduces an estimator that is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Chen et al. [4] improve on the Ledoit–Wolf method by conditioning on a sufficient statistic. Later developments of linear shrinkage covariance estimates include [2,8]. Nonlinear shrinkage estimator of the covariance matrix are studied in [16,17].

The portfolio resampling approach [20,22] is another way to handle estimation error, although there are some controversies. The resampled portfolio is more diversified than the naive MV portfolio, and this may lead to higher stability and lower transaction costs. When compared with naive MV approach, several researches recognized advantages of resampled portfolio, such as [5,6,23,25]. However, some researches show little improvement or no improvement, e.g. [1,12]. When compared with Bayesian approach, there are more evidences in favor of Bayes than resampled portfolio. Although Markowitz and Usmen [19] demonstrate resampling procedure dominates the Bayesian procedure via simulation, opposite results shows that Bayesian procedure is a better alternative, e.g. [9,18,24]. However, Michaud and Michaud [21] reply to Liechty et al. [18] that numerical implementation of the resampling procedure was less accurate. When compared with shrinkage method, the resampled portfolio is in general of less favor, and shrinkage method tends to outperform resampled portfolio method, e.g. [25,27]. A comprehensive review of the pros and cons of resampled portfolio, together with a theoretical framework, can be founded in [7].

As the rapid development in the field of financial investment, the numbers of listed stocks are increasing, asset managers needs to handle large portfolios efficiently. To allocate or manage a portfolio with 1000 stocks, there are more than 500,000 parameters to be estimated in the covariance matrix. This poses great challenge of analyzing the data, such as dimensionality and estimation error. To solve such challenge, we may consider methodologies which incorporate several effective statistical tools instead of applying them independently. In this paper, we revisit the shrinkage estimator of covariance matrix and resampling method in the context of active management. In particular, we investigate whether a combination of the two approaches could produce a better performance compared with using shrinkaged covariance alone. We propose a new algorithm to integrate linear or nonlinear shrinkage estimation with resampled efficiency, which is different from the integrated algorithm in [27]. We combined resampling method with several shrinkage estimators of covariance matrix and studied their performance. The simulation results and empirical studies show that our new algorithm could lead to a further improvement over the shrinkage estimation and Wolf's approach.

The remainder of the paper is organized as follows. In Section 2, we describe the active portfolio optimization setting and briefly review the shrinkage estimation and resampling method. In Section 3, we propose a new algorithm to combine shrinkage covariance matrix and resampling. Simulation studies are provided in Section 4, and an empirical study is given is Section 5. Conclusion and discussion are given in Section 6.

2. Shrinkage estimator versus resampling

We studies the effectiveness of shrinkage estimation and resampling techniques in the context of active management. Let $\mathbf{\Sigma }$ be the $p\times p$ covariance matrix of individual asset returns with elements ${\sigma }_{ij}$. The benckmark portfolio is ${\mathbf{w}}_B$ and the manager's portfolio is ${\mathbf{w}}_p$. The active portfolio is defined as ${\mathbf{w}}_a={\mathbf{w}}_p-{\mathbf{w}}_B$, the different holdings between the manager's portfolio and the benckmark portfolio. The benchmark portfolio is typically a weighted index of individual stocks, which is added up to unity. That is, ${\mathbf{w}}_B^{\mathrm{\prime }}\mathbf{\text{1}}=1$, where $\mathbf{\text{1}}$ is a vector of ones of length p. As we require ${\mathbf{w}}_p^{\mathrm{\prime }}\mathbf{\text{1}}=1$ for a fully invested portfolio, the active holdings add up to zero, ${\mathbf{w}}_a^{\mathrm{\prime }}\mathbf{\text{1}}=0$. The active weights are both positive and negative and expresses the views of the manager. But the manager does not have complete freedom. Due to the long-only constraint, all of the portfolio weights ${\mathbf{w}}_p$ be positive, hence ${\mathbf{w}}_a\ge -{\mathbf{w}}_B$. In addition, the manager often is faced with the constraint that the total position in any given stock cannot exceed a certain value. If this upper bound is denoted by γ, the resulting constraint on the active weights is ${\mathbf{w}}_a\le \gamma \mathbf{\text{1}}-{\mathbf{w}}_B$.

The manager's optimization problem is as follows:

$$\begin{array}{rl}\underset{\mathbf{w}}{min}& {\mathbf{w}}_a^{\mathrm{\prime }}\mathbf{\Sigma }{\mathbf{w}}_a\\ \text{subject to }& {\mathbf{w}}_a^{\mathrm{\prime }}\mathit{\mu }=g\\ & {\mathbf{w}}_a^{\mathrm{\prime }}\mathbf{\text{1}}=0\\ & {\mathbf{w}}_a\ge -{\mathbf{w}}_B\\ & {\mathbf{w}}_a\le \gamma \mathbf{\text{1}}-{\mathbf{w}}_B\end{array}$$ (1)

where g is a given excess expected return, $\mathit{\mu }$ is the $p\times 1$ vector of expected returns on all assets in the investment universe. The minimization could also be formulated as

$$\begin{array}{rl}\underset{{\mathbf{w}}_a}{max}& {\mathbf{w}}_a^{\mathrm{\prime }}\mathit{\mu }-\lambda {\mathbf{w}}_a^{\mathrm{\prime }}\mathbf{\Sigma }{\mathbf{w}}_a\\ \text{subject to }& {\mathbf{w}}_a^{\mathrm{\prime }}\mathbf{\text{1}}=0\\ & {\mathbf{w}}_a\ge -{\mathbf{w}}_B\\ & {\mathbf{w}}_a\le \gamma \mathbf{\text{1}}-{\mathbf{w}}_B\end{array}$$ (2)

where λ is the risk aversion parameter.

2.1. Shrinkage covariance

In practice, the parameters $\mathit{\mu }$ and $\mathbf{\Sigma }$ need to be estimated from history data of asset return. Let $\hat{\mathit{\mu }}$ and $\hat{\mathbf{\Sigma }}$ be the estimate, then the manager's optimization problem becomes

$$\begin{array}{rl}\underset{{\mathbf{w}}_a}{max}& {\mathbf{w}}_a^{\mathrm{\prime }}\hat{\mathit{\mu }}-\lambda {\mathbf{w}}_a^{\mathrm{\prime }}\hat{\mathbf{\Sigma }}{\mathbf{w}}_a\\ \text{subject to }& {\mathbf{w}}_a^{\mathrm{\prime }}\mathbf{\text{1}}=0\\ & {\mathbf{w}}_a\ge -{\mathbf{w}}_B\\ & {\mathbf{w}}_a\le \gamma \mathbf{\text{1}}-{\mathbf{w}}_B\end{array}$$ (3)

As $\hat{\mathit{\mu }}$ can be historical return average or a Fama-French three-factor model, we focus on the estimation of covariance. When the number of observation is smaller than the number of individual assets, using the unbiased sample covariance as an estimate of $\mathbf{\Sigma }$ could bear large estimation error. In this paper, we consider both linear and nonlinear shrinkage covariance estimation. Let $\mathbf{S}$ be the sample covariance matrix and $\mathbf{F}$ be the structured estimator. Ledoit and Wolf [13] proposed a convex linear combination between $\mathbf{S}$ and $\mathbf{F}$ as the estimation of covariance,

$${\tilde{\mathbf{\Sigma }}}_{Shrink}=\delta \mathbf{F}\mathbf{+}\mathbf{(}\mathbf{1}\mathbf{-}\delta \mathbf{)}\mathbf{S}\mathbf{,}$$ (4)

where δ is a number between 0 and 1 to determine the level of shrinkage and can be estimated to achieve optimal shrinkage. $\mathbf{F}$ could take the single-factor model covariance matrix, the identity matrix introduced in [15], and the constant correlation model proposed by [14]. In addition, we consider the optimal nonlinear shrinkage of large-dimensional covariance matrices proposed in [17].

2.2. Portfolio resampling

Resampling technique proposed by Michaud and Michaud [22] is to avoid drawback of relying only on the sample covariance matrix $\mathbf{S}$ computed from past returns. The specific process is as follows. Suppose we have the T observational returns of p assets, we estimate the $p\times p$ variance–covariance matrix ${\hat{\mathbf{\Sigma }}}_0$ and the $p\times 1$ expected returns vectors ${\hat{\mathit{\mu }}}_0$. Portfolio resampling is a parametric bootstrap procedure which draws sample from the estimated return distribution with sample size T. By repeating the sampling procedure n times, we get n new sets of mean vector and covariance matrices $({\hat{\mathbf{\Sigma }}}_1,{\hat{\mathit{\mu }}}_1),\dots ,({\hat{\mathbf{\Sigma }}}_n,{\hat{\mathit{\mu }}}_n)$. For each set, we can calculate m portfolios evenly along the frontier spanning from the minimum-variance portfolio to the maximum-return portfolio, then obtain the corresponding weights for each n, ${\mathbf{w}}_{11},\dots ,{\mathbf{w}}_{1m}$ to ${\mathbf{w}}_{n1},\dots ,{\mathbf{w}}_{nm}$. The weight of a resampled portfolio is

$${\mathbf{w}}_j^{re}=\frac{1}{n}\sum _{i=1}^n{\mathbf{w}}_{ij}j=1,\dots ,m,$$ (5)

where ${\mathbf{w}}_{ij}$ denotes the $p\times 1$ vector of the jth portfolio along the frontier for the ith resampling.

Scherer [23] showed that the unstable performance of resampling is close to the sample covariance matrix in the absence of lower and upper bounds on portfolio weights. However, imposing lower and upper bounds on portfolio weights lead to more diversified portfolios compared to the sample covariance matrix, and resampling does improve out-of-sample performance.

3. Combining shrinkage and resampling

In the presence of a long-only constraint, both resampling and shrinkage estimation improve upon the sample covariance matrix. Wolf [27] conclude that although both shrinkage and resampling improves upon the sample covariance matrix, shrinkage is superior than resampling. In addition, Wolf [27] describes an algorithm of combining the two methods. The details of the algorithm is provided in below.

Algorithm 3.1 Wolf's resampling with shrinkage, REA1 —

 

1. Resample from the past returns using parametric bootstrap to obtain a bootstrap sequence of return.

2. Compute the shrinkage estimator from the bootstrap data and call it ${\hat{\mathbf{\Sigma }}}_{Shrink}$.

3. Solve the quadratic optimization problem (1). Let ${\mathbf{w}}_a^{\ast }$ be the resulting optimal vector of active weights.

4. Repeat the first three steps n times and average over the n active weight vectors ${\mathbf{w}}_a^{\ast }$ to obtain the final vector of active portfolio weights ${\tilde{\mathbf{w}}}_a^{\ast }$.

The empirical analysis in [27] demonstrate that resampling combined with shrinkage in Algorithm 3.1 does not offers any further improvement on ‘pure’ shrinkage. The seminal paper [10] proved that constrained MV solution with lower and upper bounds is equivalent to unconstrained MV solution with a shrunked covariance, Therefore, resampling with lower and upper bounds is equivalent to a combination of resampling and covariance shrinkage. This conclusion helps to explain the results in [27]. A parametric bootstrap implementation of Algorithm 3.1 resample from sample covariance, and a shrinkage estimator based on the bootstrap sample is used for MV optimization. Since optimization contains lower and upper bounds corresponds to a further step of covariance shrinkage, the contribution of step 2 is difficult to assess and may not have significant function.

In light of the above analysis, we propose a better algorithm to combine the two methods, in which we resample from a shrinkage covariance. The algorithm is described as follows:

Algorithm 3.2 Resampling after shrinkage, REA2 —

1. Compute the shrinkage estimator from the past data as ${\hat{\mathbf{\Sigma }}}_{Shrink}$.

2. Resampling from a model with ${\hat{\mathbf{\Sigma }}}_{Shrink}$ to obtain a sequence of bootstrap return.

3. Compute the sample covariance matrix from the resampled data and call it $\tilde{\mathbf{\Sigma }}$.

4. Solve the quadratic optimization problem (3) with $\tilde{\mathbf{\Sigma }}$. Let ${\mathbf{w}}_a^{\ast }$ be the resulting optimal vector of active weights.

5. Repeat steps $2,3\text{ and }4$ n times and average over the n active weight vectors ${\mathbf{w}}_a^{\ast }$ to obtain the final vector of active portfolio weights ${\tilde{\mathbf{w}}}_a^{\ast }$.

Next, we illustrate the proposed method via simulation and empirical study and compare with pure shrinkage and Algorithm 3.1. The results indicate that our resampling with shrinkage improvement over Algorithm 3.1 as well as the ‘pure’ shrinkage method.

4. Simulation study

In this section, we conduct simulation studies to demonstrate our method and compare the effectiveness of resampling, shrinkage estimation and their combinations. In order to generate simulation settings close to reality, we use the monthly returns of industrial value-weighted portfolios from the website of Kenneth French from January 1998 to October 2018. For each simulation we bootstrap a sample of size 120 from the real data, which contains 49 assets.An equal-weighted index is served as our benchmarks. We estimated the expected return $\hat{\mathit{\mu }}$, the estimated covariance matrices $\hat{\mathbf{\Sigma }}s$ using the past 60 months return data, and set an upper bound of $\gamma =0.1$ as the largest weight of any asset. The expected return for the next month is also estimated using the Fama-French three-factor model. The covariance matrix of the 49 assets is estimated according to five estimators, sample covariance matrix, the shrinkage estimation of $\mathbf{F}$ is an identity matrix, denoted by ‘Shrink-I’; the shrinkage estimation of $\mathbf{F}$ is Single-Factor risk model, denoted by ‘Shrink-SF’; the shrinkage estimation of $\mathbf{F}$ is a constant correlation covariance matrix, denoted by ‘Shrink-CC’; and the nonlinear shrinkage estimation, denoted by ‘Shrink-NS’. For MV optimization, the quadratic optimizer output the weight vectors ${\hat{\mathbf{w}}}_a$. The resampling method computes weight vectors ${\tilde{\mathbf{w}}}_a$. At the end of the month, the realized excess return is recorded as $\alpha ={\hat{\mathbf{w}}}_a^{\mathrm{\prime }}\mathbf{r}$ for naive MV, and as ${\alpha }^{\ast }={\tilde{\mathbf{w}}}_a^{\mathrm{\prime }}\mathbf{r}$ for resampling, where $\mathbf{r}$ is the vector of stock return for the month. The portfolios are then held for one month and rebalanced at the beginning of the next month.

The results of 100 simulations are shown in Tables 1–3. The resampling portfolio is denoted by ‘RE’, the Wolf's resampling with shrinkage is denoted by ‘REA1’, and our proposed resampling with shrinkage is denoted by ‘REA2’. We consider various risk aversion parameter $\lambda =0.1$, 1.0, 10.0, 100.0, and 10000.0. The 60 realized excess return are obtained for each λ in the simulation. From the excess returns we compute the annualized realized information ratio as $\sqrt{12}\overline{\alpha }/{\sigma }_{\alpha }$, where $\overline{\alpha }$ is the sample average of the excess returns and ${\sigma }_{\alpha }$ is the sample standard deviation of the excess returns.

Table 1. This table presents average (std) annualized excess returns, risk, and ex post information ratios of the portfolios of the MV optimized by the sample covariance matrix and several shrinkage estimators.

	λ	0.1	1.0	10.0	100.0	10000.0
	α (%)	20.38 (3.19)	12.23 (1.90)	2.09 (0.69)	0.19 (0.23)	0.00 (0.00)
Sample cov	${\sigma }_a$ (%)	7.91 (1.09)	5.36 (0.96)	1.80 (0.29)	0.49 (0.08)	0.01 (0.00)
	IR	2.59 (0.29)	2.31 (0.32)	1.16 (0.33)	0.36 (0.43)	0.31 (0.43)
	α (%)	20.47 (3.24)	12.97 (1.90)	2.09 (0.41)	0.21 (0.04)	0.00 (0.00)
Shrink-I	${\sigma }_a$ (%)	7.93 (1.09)	5.47 (1.00)	1.12 (0.30)	0.11 (0.03)	0.00 (0.00)
	IR	2.59 (0.29)	2.40 (0.32)	1.94 (0.37)	1.93 (0.37)	1.93 (0.37)
	α (%)	20.47 (3.27)	14.48 (1.88)	2.80 (0.57)	0.29 (0.07)	0.00 (0.00)
Shrink-SF	${\sigma }_a$ (%)	7.96 (1.07)	5.86 (1.03)	1.44 (0.50)	0.15 (0.06)	0.00 (0.00)
	IR	2.58 (0.30)	2.51 (0.31)	2.05 (0.36)	2.00 (0.38)	2.00 (0.38)
	α (%)	20.65 (3.26)	14.83 (2.35)	2.76 (0.58)	0.28 (0.06)	0.00 (0.00)
Shrink-CC	${\sigma }_a$ (%)	7.94 (1.08)	5.93 (1.03)	1.38 (0.39)	0.14 (0.04)	0.00 (0.00)
	IR	2.61 (0.30)	2.53 (0.33)	2.06 (0.36)	2.04 (0.36)	2.04 (0.36)
	α (%)	20.40 (3.26)	13.03 (1.92)	2.15 (0.43)	0.22 (0.05)	0.00 (0.00)
Shrink-NS	${\sigma }_a$ (%)	7.93 (1.09)	5.50 (0.99)	1.16 (0.31)	0.12 (0.03)	0.00 (0.00)
	IR	2.58 (0.29)	2.40 (0.30)	1.90 (0.33)	1.88 (0.33)	1.88 (0.33)

Note: Standard deviation in parentheses.

Table 3. Average (std) monthly turnover.

	λ	0.1	1.0	10.0	100.0	10000.0
	Sample cov	0.68 (0.03)	0.52 (0.03)	0.19 (0.01)	0.05 (0.00)	0.02 (0.00)
	Shrink-I	0.68 (0.03)	0.51 (0.03)	0.09 (0.01)	0.02 (0.00)	0.02 (0.00)
MV	Shrink-SF	0.69 (0.03)	0.54 (0.03)	0.12 (0.02)	0.02 (0.00)	0.02 (0.00)
	Shrink-CC	0.69 (0.03)	0.54 (0.02)	0.11 (0.01)	0.02 (0.00)	0.02 (0.00)
	Shrink-NS	0.68 (0.03)	0.51 (0.03)	0.10 (0.01)	0.02 (0.00)	0.02 (0.00)
	Sample cov	0.68 (0.03)	0.51 (0.03)	0.23 (0.01)	0.10 (0.01)	0.02 (0.00)
	Shrink-I	0.68 (0.03)	0.51 (0.03)	0.14 (0.02)	0.02 (0.00)	0.02 (0.00)
REA1	Shrink-SF	0.68 (0.03)	0.53 (0.03)	0.15 (0.02)	0.03 (0.00)	0.02 (0.00)
	Shrink-CC	0.68 (0.03)	0.53 (0.03)	0.15 (0.02)	0.03 (0.00)	0.02 (0.00)
	Shrink-NS	0.68 (0.03)	0.51 (0.03)	0.17 (0.01)	0.03 (0.00)	0.02 (0.00)
	Sample cov	0.68 (0.03)	0.51 (0.03)	0.23 (0.01)	0.10 (0.01)	0.02 (0.00)
	Shrink-I	0.68 (0.03)	0.51 (0.03)	0.19 (0.02)	0.05 (0.00)	0.02 (0.00)
REA2	Shrink-SF	0.68 (0.03)	0.54 (0.03)	0.22 (0.02)	0.06 (0.01)	0.02 (0.00)
	Shrink-CC	0.68 (0.03)	0.54 (0.02)	0.22 (0.02)	0.06 (0.01)	0.02 (0.00)
	Shrink-NS	0.68 (0.03)	0.51 (0.03)	0.20 (0.02)	0.06 (0.00)	0.02 (0.00)

From Table 1, it can be seen that shrinkage improves upon the sample covariance matrix for larger λ. A larger λ suggests a lower level of confidence in predicting the expected return, the optimization is mainly based on the estimated covariance matrix. The result illustrates that the several shrinkage estimators are better than the sample covariance matrix. This agrees with previous researches.

From Table 2, we can conclude that both Wolf's REA1 algorithm and our REA2 algorithm are better than resampling method with the sample covariance matrix. In terms of information ratio, our REA2 algorithm is superior to Wolf's REA1 algorithm except in the setting of $\lambda =0.1$, and $\lambda =10000$, though Wolf's REA1 algorithm has slightly smaller variance. Comparing Tables 1 and 2, it can be concluded that resampling method RE is superior to the sample covariance matrix but does not superior to shrinkage method. That is, resampling overcomes the estimation error of the covariance matrix to some extent. In addition, Wolf's resampling combined with shrinkage (REA1) does not improve the ‘pure’ shrinkage, however, our proposed REA2 improves the ‘pure’ shrinkage in the settings of when λ takes values of 1, 10, and 100.

Table 2. This table presents average (std) annualized returns, risk, and ex post information ratios of the resampling portfolios constructed using the sample covariance matrix and several shrinkage estimators.

		λ	0.1	1.0	10.0	100.0	10000.0
		α (%)	20.35 (3.22)	13.15 (1.88)	3.66 (0.75)	0.73 (0.35)	0.01 (0.01)
Sample cov	RE	${\sigma }_a$ (%)	7.84 (1.09)	5.36 (0.93)	2.09 (0.36)	0.87 (0.10)	0.03 (0.01)
		IR	2.61 (0.30)	2.48 (0.32)	1.76 (0.32)	0.83 (0.38)	0.30 (0.43)
Shrink-I		α (%)	20.40 (3.24)	13.55 (1.89)	3.16 (0.58)	0.34 (0.07)	0.00 (0.00)
	REA1	${\sigma }_a$ (%)	7.87 (1.09)	5.48 (0.94)	1.61 (0.39)	0.18 (0.05)	0.00 (0.00)
		IR	2.60 (0.30)	2.50 (0.31)	2.01 (0.33)	1.92 (0.35)	1.92 (0.35)
		α (%)	20.42 (3.25)	13.86 (1.96)	4.39 (0.75)	1.01 (0.20)	0.01 (0.00)
	REA2	${\sigma }_a$ (%)	7.88 (1.09)	5.49 (0.92)	1.99 (0.44)	0.50 (0.12)	0.01 (0.00)
		IR	2.60 (0.30)	2.55 (0.31)	2.26 (0.37)	2.05 (0.40)	1.89 (0.39)
Shrink-SF		α (%)	20.44 (3.26)	14.16 (1.95)	3.44 (0.57)	0.38 (0.07)	0.00 (0.00)
	REA1	${\sigma }_a$ (%)	7.89 (1.08)	5.61 (0.98)	1.67 (0.42)	0.20 (0.06)	0.00 (0.00)
		IR	2.60 (0.30)	2.56 (0.32)	2.12 (0.33)	2.02 (0.35)	2.02 (0.35)
		α (%)	20.46 (3.27)	15.06 (1.97)	5.32 (0.71)	1.30 (0.23)	0.02 (0.00)
	REA2	${\sigma }_a$ (%)	7.91 (1.08)	5.83 (0.99)	2.29 (0.52)	0.61 (0.17)	0.01 (0.00)
		IR	2.60 (0.30)	2.61 (0.31)	2.39 (0.35)	2.20 (0.35)	1.98 (0.38)
Shrink-CC		α (%)	20.51 (3.26)	14.37 (1.99)	3.43 (0.57)	0.37 (0.07)	0.00 (0.00)
	REA1	${\sigma }_a$ (%)	7.89 (1.09)	5.68 (0.94)	1.66 (0.41)	0.19 (0.06)	0.00 (0.00)
		IR	2.61 (0.30)	2.56 (0.31)	2.13 (0.35)	2.04 (0.37)	2.04 (0.37)
		α (%)	20.60 (3.29)	15.36 (2.30)	5.41 (0.99)	1.32 (0.27)	0.02 (0.00)
	REA2	${\sigma }_a$ (%)	7.90 (1.09)	5.89 (0.98)	2.29 (0.51)	0.60 (0.15)	0.01 (0.00)
		IR	2.62 (0.30)	2.64 (0.32)	2.41 (0.37)	2.23 (0.38)	2.02 (0.40)
Shrink-NS		α (%)	20.37 (3.25)	13.53 (1.90)	3.14 (0.60)	0.36 (0.10)	0.00 (0.00)
	REA1	${\sigma }_a$ (%)	7.87 (1.08)	5.50 (0.94)	1.74 (0.37)	0.25 (0.05)	0.00 (0.00)
		IR	2.60 (0.30)	2.49 (0.32)	1.83 (0.32)	1.44 (0.30)	1.42 (0.30)
		α (%)	20.38 (3.29)	13.85 (1.92)	4.44 (0.69)	1.03 (0.19)	0.01 (0.00)
	REA2	${\sigma }_a$ (%)	7.87 (1.08)	5.53 (0.94)	2.03 (0.43)	0.52 (0.12)	0.01 (0.00)
		IR	2.60 (0.30)	2.53 (0.30)	2.23 (0.34)	2.02 (0.34)	1.85 (0.32)

Note: ‘REA1’ represents wolf's resampling combined with shrinkage algorithm, and ‘REA2’ represents our shrinkage combined with resampling algorithm.

Portfolio turnover is the changes in portfolios weights. If the targeted weights are ${\mathbf{w}}^{new}=(w_1^{new},\dots ,w_p^{new}{)}^{\mathrm{\prime }}$, and the current portfolio weights are ${\mathbf{w}}^{old}=(w_1^{old},\dots ,w_p^{old}{)}^{\mathrm{\prime }}$, then the turnover for the portfolio update can be calculated as

$$T=\frac{1}{2}\sum _{i=1}^p|w_i^{new}-w_i^{old}|.$$

In general, portfolio manager prefer a smaller turnover to save transaction cost. Table 3 present statistics on the average monthly turnover. According to the results of Table 3, we draw the following 3 conclusions based on our simulation. First, for larger λ, say 10.0, 100.0, 10000.0, the ‘pure’ resampling has the highest turnover. Second, shrinkage improves upon the sample covariance matrix. Third, both REA1 and REA2 improve upon ‘pure’ resampling but does not improve upon ‘pure’ shrinkage.

5. Empirical application

In this section, we study out-of-sample performance of our method and make comparisons with shrinkage estimation, resampling and Wolf's approach. We use the monthly returns of 100 most liquid stocks in the $S\mathrm{\&}P$ 500 from January 2000 to February 2017. An equal-weighted index is served as a benchmarks. Before optimization, the expected return for the next month is estimated using the Fama-French three-factor model obtained from the past 120 months data, and the covariance matrix of the 100 stocks is estimated according to the five estimators using the ‘past’ 120 months return data. At the beginning of each month from 2010 to 2017, we construct an optimal portfolio, and the portfolio is held for one month and rebalanced at the beginning of the next month. The out-of-sample period ranges from January 2010 until February 2017, hence there are 86 months realized excess returns obtained for each λ. From the excess returns, we compute the annualized average excess return, risk of excess returns, and the realized information ratio as in simulation.

Summary statistics for the realized excess returns are presented in Tables 4 and 5. From Table 4, we can get that shrinkage improves upon the sample covariance matrix. From Table 5, we can conclude that our REA2 algorithm is superior to Wolf's REA1 algorithm except for the case $\lambda =0.1$. While Wolf's REA1 algorithm shows little or no improvement to the ‘pure’ shrinkage method, our REA2 algorithm is able to improve upon the ‘pure’ shrinkage method in several settings, e.g. when λ takes values of 1, 10, and 100.

Table 4. This table presents annualized returns, annualized risk, and ex post information ratios of the portfolios of the MV optimized by the sample covariance matrix and several shrinkage estimators.

	λ	0.1	1.0	10.0	100.0	10000.0
	$\overline{\alpha }$ (%)	12.4034	5.9753	0.8754	−0.0120	−0.0005
Sample cov	Tracking error(%)	8.7471	4.2933	1.4101	0.4899	0.0052
	IR	1.4180	1.3918	0.6208	−0.0245	−0.0971
	$\overline{\alpha }$ (%)	12.0275	6.2761	0.8058	0.0806	0.0008
Shrink-I	Tracking error(%)	8.8353	4.0811	0.6428	0.0643	0.0006
	IR	1.3613	1.5378	1.2536	1.2537	1.2537
	$\overline{\alpha }$ (%)	10.8542	6.4213	0.8977	0.0897	0.0009
Shrink-SF	Tracking error(%)	8.8725	4.1211	0.6166	0.0615	0.0006
	IR	1.2234	1.5582	1.4560	1.4586	1.4586
	$\overline{\alpha }$ (%)	10.9290	8.0239	1.2256	0.1221	0.0012
Shrink-CC	Tracking error(%)	8.7249	4.5904	0.7069	0.0699	0.0007
	IR	1.2526	1.7480	1.7338	1.7476	1.7476
	$\overline{\alpha }$ (%)	12.0582	6.2594	0.7981	0.0798	0.0008
Shrink-NS	Tracking error(%)	8.9376	4.0262	0.5915	0.0591	0.0006
	IR	1.3491	1.5547	1.3494	1.3495	1.3495

Note: The out-of-sample period is 01/2010 until 02/2017, yield 86 monthly excess returns.

Table 5. This table presents annualized characteristics of the portfolios of the two resampling optimization algorithms.

		λ	0.1	1.0	10.0	100.0	10000.0
Sample cov		$\overline{\alpha }$ (%)	12.2508	7.0676	1.6740	0.3175	−0.0038
	RE	Tracking error(%)	8.6302	4.4189	1.6781	0.8167	0.0328
		IR	1.4195	1.5994	0.9975	0.3888	−0.1166
Shrink-I		$\overline{\alpha }$ (%)	12.1129	7.1723	1.3647	0.1426	0.0014
	REA1	Tracking error(%)	8.6907	4.3869	1.1404	0.1228	0.0012
		IR	1.3938	1.6349	1.1966	1.1606	1.1606
		$\overline{\alpha }$ (%)	11.8182	7.3904	1.8393	0.4339	0.0050
	REA2	Tracking error(%)	8.6596	4.4222	1.3646	0.3475	0.0040
		IR	1.3648	1.6712	1.3478	1.2485	1.2667
Shrink-SF		$\overline{\alpha }$ (%)	11.6827	7.3336	1.3799	0.1423	0.0014
	REA1	Tracking error(%)	8.7308	4.3243	0.9361	0.0967	0.0010
		IR	1.3381	1.6959	1.4742	1.4721	1.4721
		$\overline{\alpha }$ (%)	11.0245	7.4409	2.0556	0.4871	0.0057
	REA2	Tracking error(%)	8.7639	4.4609	1.3387	0.3351	0.0040
		IR	1.2580	1.6680	1.5355	1.4536	1.4222
Shrink-CC		$\overline{\alpha }$ (%)	11.7287	7.7323	1.4104	0.1451	0.0015
	REA1	Tracking error(%)	8.6610	4.4799	0.9262	0.0947	0.0009
		IR	1.3542	1.7260	1.5227	1.5327	1.5327
		$\overline{\alpha }$ (%)	11.0514	8.9709	2.7191	0.6444	0.0074
	REA2	Tracking error(%)	8.6114	4.9135	1.5074	0.3672	0.0043
		IR	1.2833	1.8258	1.8038	1.7548	1.7371
Shrink-NS		$\overline{\alpha }$ (%)	11.9061	7.3071	1.2877	0.1163	0.0012
	REA1	Tracking error(%)	8.7676	4.3242	1.1355	0.1421	0.0014
		IR	1.3580	1.6898	1.1340	0.8180	0.8163
		$\overline{\alpha }$ (%)	11.7751	7.4434	1.9611	0.4242	0.0047
	REA2	Tracking error(%)	8.7495	4.3741	1.3580	0.3322	0.0038
		IR	1.3458	1.7017	1.4442	1.2768	1.2577

Note: The out-of-sample period is 01/2010 until 02/2017, yield 86 monthly excess returns.

Table 6 present statistics on the average monthly turnover. The conclusion is similar to the simulation on turnover. The ‘pure’ resampling has the highest turnover, and ‘pure’ shrinkage has the lowest turnover. Both REA1 and REA2 improve upon ‘pure’ resampling but does not improve upon ‘pure’ shrinkage.

Table 6. Average monthly turnover.

	λ	0.1	1.0	10.0	100.0	10000.0
	Sample cov	0.7635	0.5319	0.1764	0.0529	0.0210
	Shrink-I	0.7616	0.4936	0.0881	0.0233	0.0210
MV	Shrink-SF	0.7626	0.5206	0.0960	0.0241	0.0210
	Shrink-CC	0.7693	0.5483	0.1004	0.0245	0.0210
	Shrink-NS	0.7598	0.4920	0.0843	0.0232	0.0210
	Sample cov	0.7543	0.5264	0.2225	0.0927	0.0212
	Shrink-I	0.7550	0.5151	0.1531	0.0279	0.0210
REA1	Shrink-SF	0.7569	0.5204	0.1320	0.0266	0.0210
	Shrink-CC	0.7607	0.5272	0.1303	0.0265	0.0210
	Shrink-NS	0.7545	0.5101	0.1451	0.0291	0.0210
	Sample cov	0.7543	0.5264	0.2225	0.0927	0.0212
	Shrink-I	0.7537	0.5094	0.1867	0.0549	0.0210
REA2	Shrink-SF	0.7569	0.5331	0.2015	0.0586	0.0210
	Shrink-CC	0.7636	0.5604	0.2119	0.0606	0.0210
	Shrink-NS	0.7530	0.5090	0.1877	0.0533	0.0210

6. Conclusion and discussion

Extensive researches focus on the estimation error in MV approach for portfolio optimization, and two methods are studied in literature, the shrinkage method and the resampling method. While it is recolonized that shrinkage method does improve the performance, there are some debates on the resampling method. The resampling technique is closely related to the bootstrap aggregating method [3] and can be viewed as a Bayesian posterior mean estimation with a noninformative prior [26]. The effectiveness of resampling depends on whether such a prior is appropriate.

In this paper, we compare the performance of resampling, shrinkage estimation and their combinations. Our analysis from both simulation and empirical analysis shows that shrinkage method improve over the naive MV approach, give better performance than resampling. In addition, Furthermore, we find evidence that a combination of the two method in the REA2 algorithm offers further improvement over and ‘pure’ shrinkage method, and better than the Wolf's REA1 combination algorithm.

Disclosure statement

No potential conflict of interest was reported by the authors.