Portfolio Selection via Shrinkage by Cross Validation
Given the importance of the loss function choice [Christoffersen, P. and K. Jacobs (2004) The importance of the loss function in option valuation. J. Financial Economics 72: 291-318], this paper proposes the nonparametric technique of cross validation, to tuning the shrinkage intensity estimation of Ledoit and Wolf [Ledoit, O. and W. Michael (2003) Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J. Empirical Finance 10: 603-621; Ledoit, O. and W. Michael (2004) Honey, I Shrunk the Sample Covariance Matrix. J. Portfolio Management 30: 110-119; Ledoit, O. and W. Michael (2004) A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Analysis 88: 365-411]. By aligning the loss function of out-of-sample forecast identical to the one used for the shrinkage intensity estimation, the proposed cross validation approach shows the significant gains in terms of both the variance reduction and information ratio improvement to various portfolios of the U.S. firms.
At a glance: Figures
Keywords: cross validation, shrinkage targeting, portfolio choice, shrinkage intensity, loss function
Journal of Finance and Accounting, 2014 2 (4),
Received May 18, 2014; Revised July 10, 2014; Accepted July 15, 2014Copyright © 2014 Science and Education Publishing. All Rights Reserved.
Cite this article:
- Liu, Xiaochun. "Portfolio Selection via Shrinkage by Cross Validation." Journal of Finance and Accounting 2.4 (2014): 74-81.
- Liu, X. (2014). Portfolio Selection via Shrinkage by Cross Validation. Journal of Finance and Accounting, 2(4), 74-81.
- Liu, Xiaochun. "Portfolio Selection via Shrinkage by Cross Validation." Journal of Finance and Accounting 2, no. 4 (2014): 74-81.
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Variance-covariance estimation is a fundamental issue to portfolio selection. The estimation approximated by the sample counterpart brings large estimation errors to high dimensional portfolios, the variance-covariance of which is ill-conditioned in general. Deriving solid theoretical assumptions and asymptotic solutions, Ledoit and Wolf (2003, 2004a, 2004b) apply the shrinkage method to estimate a well-conditioned estimator of a large-dimensional covariance matrix.
Among their solutions, determining shrinkage intensity is crucial to out-of-sample portfolio performance. They use the shrinkage weights estimated by in-sample periods to select portfolios in out-of-sample. However, this procedure raises a critical issue as to the choice of loss functions. The importance of the aligned loss function in model estimation and evaluation has been emphasized by Christoffersen and Jacobs (2004). They show that the loss function should be identical for the estimation and evaluation of out-of-sample forecasts; otherwise inappropriate conclusions will be made. In fact, it can be argued that the choice of loss function implicitly defines the model under consideration (Engle, 1993). Moreover, I postulate that it will generally be preferable to estimate the parameters for such an out-of-sample exercise using an identical in-sample estimation loss function. Additionally, Bams et al (2009) confirm the empirical results of Christoffersen and Jacobs (2004), and provide a method to evaluate the adequacy of the loss function.
This paper proposes the nonparametric technique of cross validation to align the out-of-sample forecast loss function for a portfolio estimated by covariance shrinkage. The cross validation technique splits a sample dataset into the training and validating periods. The parameters of portfolio selection models are estimated from the training periods, and then evaluated by the validating period. The validating period is an analog to an out-of-sample period. The parameters with the best performance evaluated in the validating period will be chosen for out-of-sample forecasts. Such a validation to out-of-sample forecast is expected to mitigate data snooping and over-fitting issues, which generally exist in the out-of-sample estimations. The validating period is the key to align the loss functions of out-of-sample forecasts. To the best of my knowledge, this paper is the first to use cross validation in portfolio selection of high dimension through covariance shrinkage. Most importantly, cross validation can align loss functions between estimation and evaluation of out-of-sample portfolios.
Particularly, this paper adopts the leave-one-out cross validation with three criteria to tuning the shrinkage intensity parameters for mean-variance and global minimum variance portfolios. The portfolios with different sizes and expected returns estimated by cross validation significantly outperform those of Ledoit and Wolf in terms of both the variance reduction and the information ratio improvement. Furthermore, the cross validation portfolios have lower turnovers which are more attractive to an active manager. Additionally, the shrinkage intensity estimated by cross validation is more volatile and higher than those of Ledoit and Wolf. As a result, this might imply that the cross validation estimation quickly adjusts to monthly updated market information.
This paper is organized as follows. Section 2 reviews the literature. Section3 constructs portfolios and presents optimal shrinkage intensity. Section 4 describes the cross validation procedure for estimating optimal shrinkage intensity. Section 5 describes data. Section 6 reports empirical results. Section 7 concludes the paper.
The cross validation technique is not new in statistics literature. Used to estimate the risk of an estimator or to perform model selection, cross-validation is a widespread strategy because of its simplicity and its (apparent) universality. However, the application of cross validation in economics and finance is rare. One phenomenal exception is Conway and Reinganum (1988) who consider the use of cross-validation to identify a stable factor structure in security returns. Fan et al. (2010) propose a two-stage refitted cross validation to attenuate the influence of irrelevant variables with high spurious correlations. Zhang (2009) designs a cross validation estimate of quadratic variation to study the nonparametric connection between realized and implied volatilities. Picard and Cook (1984) apply cross validation for assessment of the predictive ability of regression models. Upton (1994) extends the cross validation technique of Conway and Reinganum (1988) to examine both the statistical and economic significance of extracted factors. Fan and Yim (2004) use cross validation as a data-driven method to estimate conditional densities. Arlot and Celisse (2010) provide a recent survey of cross validation.
Shrinkage estimators have been widely studied to diminish the estimation error on the estimation of the moments of asset returns. DeMiguel et. al. (2011) study shrinkage estimators for the vector of means, the covariance matrix and portfolio weights themselves. Candelon et. al. (2010) introduce a new framework based on shrinkage estimators to improve the performance of small size portfolios. Golosnoy and Okhrin (2009) derive the flexible shrinkage estimator for the optimal portfolio weights, which allows dynamic adjustments of model structure. Golosnoy and Okhrin (2007) propose directly applying shrinkage to the portfolio weights by using the non-stochastic target vector. Ledoit and Wolf (2010) extends the shrinkage method by considering nonlinear transformations of the sample Eigen values, using recent results from Random Matrix Theory (RMT). Behr et al (2010) impose the set of constraints that yields the optimal trade-off between sampling error reduction and bias for the variance-covariance matrix. Kourtis et. al. (2011) apply shrinkage method directly to the inverse covariance matrix using two non-parametric methods.
The alternative methods to shrinkage approach are also studied and compared in the literature. Jagannathan and Ma (2003) show that constraining portfolio weights to be nonnegative is equivalent to using the sample covariance matrix after reducing its large elements and then form the optimal portfolio without any restrictions on portfolio weights. DeMiguel et al. (2009) solve the traditional minimum-variance problem subject to the additional constraint that the norm of the portfolio-weight vector be smaller than a given threshold. Disatnik and Benninga (2007) show that there is no real need to use the shrinkage estimators. Liu and Lin (2010) compare portfolio performance among different shrinkage methods. Briner and Connor (2008) find that the factor model perform best for large investment universes and typical sample lengths. Wolf (2006) compares shrinkage estimation to resampled efficiency.
3. Portfolio and Optimal Shrinkage Intensity
Consider a general mean-variance portfolio (MVP) of Markowitz (1952) type with a universe of N stocks, whose returns are distributed with mean vector , and covariance matrix, . Markowitz (1952) defines the problem of portfolio selection as:
where 1 denotes a conformable vector of ones, and q is the expected rate of return that is required on the portfolio. w is a vector of portfolio weights to be solved. The well-known solution is
with , and .
In this paper, I also estimate a global minimum variance portfolio (GMVP) as
with its solution as
Note that the solutions of both equations (1) and (2) involve the inverse of the covariance matrix. The conventional approach is to use the sample covariance matrix, , to approximate the population matrix, . However, in a high dimensional portfolio selection problem, the sample covariance is typically not well-conditioned and may not even be invertible. Ledoit and Wolf (2004b) apply the shrinkage method to obtain an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. The estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret.
Let be the shrinkage estimator, and F a shrinkage targeting matrix. The shrinkage estimator is expressed as a weighted average of a shrinkage targeting matrix and the sample covariance matrix.
where is the shrinkage intensity. The shrinkage intensity reflects the trade-off of estimation errors and bias.
The optimal shrinkage intensity is given by
where , , and are the elements of , and , respectively, with , and . The asymptotically consistent estimator of the optimal shrinkage intensity has been derived in Ledoit and Wolf (2003, 2004a &b). I refer the interested readers to their works for the technical details.
4. Cross Validation for Optimal Shrinkage Intensity
The key of the shrinkage approach is to determine the shrinkage intensity () of a structure targeting matrix and the sample covariance matrix. To calibrate the shrinkage weight, I adopt the nonparametric technique, known as the leave-one-out cross validation, on a grid search of the shrinkage intensity domain. Leave-one-out, e.g., Celisse and Robin (2008), Shao (1993), Efron and Gong (1983), Campbell et al. (1997), Stone (1974), Allen (1974), Geisser (1975), is the most classical exhaustive cross validation procedure in the statistics literature.
In this paper , the procedure of leave-one-out cross validation is implemented as follows:
1. Give an estimation window of length, R, and a out-of-sample forecast window of length, H , with the total observations, T , such that .
2. Each data point at is successively “left out” from the sample and used for validation. Hence, at each data point, the training period is the set of , while the validating period is the data point.
3. The domain of is divided to a grid of D points, such that . The accuracy of the shrinkage intensity will depend on the number of grid points. Considering the computation cost, I choose D=200 in the empirical section of this paper.
4. Apply each of the grid points to equation (3) for the training period, , so as to obtain the shrinkage covariance matrix, , d=1,...D.
5. Use to compute the corresponding portfolio weights, through equation (1) and (2).
6. Compute the out-of-sample portfolio return on the sample asset return of the grid point, d , to have , where is the returns of N stocks at time .
7. Compute the out-of-sample statistics for each grid point, using the out-of-sample portfolio returns, .
8. Choose the parameter, , which optimizes the out-of-sample statistics, as the optimal shrinkage intensity.
9. Use to compute the shrinkage covariance matrix, and portfolio weight, to obtain the out-of-sample forecast of the portfolio return at time R+1 as
10. Repeat the steps (2)-(9) to obtain the out-of-sample portfolio return forecasts, , for H=R+1,...T.
As seen, the cross validation is a nonparametric approach by which a validation loss function from leave-one-out is aligned to the loss function of out-of-sample forecast. I expect that this cross validation approach by aligning loss functions will reduce the issues of data snooping and over-fitting in out-of-sample forecast.
Next, I specify the out-of-sample statistics, . In this paper, I use three different criteria for to calibrate the shrinkage intensity by: (1) minimizing out-of-sample portfolio variance (MinVariance); (2) maximizing out-of-sample portfolio Information Ratio (MaxIR); (3) minimizing out-of-sample deviation of a tracking portfolio from required expected returns (MinDeviation).
For MinVairance, the variance of the out-of-sample portfolio returns is computed on the out-of-sample portfolio returns, , obtained from leave-one-out cross validation; that is:
where . Such that,
For MaxIR, the criteria of maximizing information ratio is defined as
For MinDeviation, I compute the deviation of the out-of-sample portfolio returns from the required expected rate of return, q , in a mean-variance portfolio; thus, the out-of-sample statistics, is defined as:
See Lamont (2001) for more details in economic tracking portfolios. Finally, the determined is used in equation (3) to estimate the portfolio returns for the out-of-sample forecast period, .
Monthly U.S. stock returns, from January 1980 to December 2010, were taken from the Center for Research in Security Prices (CRSP). The sample period has the observations, T=372. The estimation window is set to R=120 such that the out-of-sample forecast periods is H=252. At each time t, I form the portfolio with the size of n U.S. stocks. The n stocks with the largest market capitalization are chosen as the portfolio stocks for time t. The sample covariance matrix is estimated based on the past R months, back from time t, of the corresponding portfolio stocks’ returns. The same procedure is repeated for every month of the out-of-sample periods from January 1990 to December 2010. As a result, the portfolios are monthly updated and rebalanced.
Kirby and Ostdiek (2011) have shown that targeting conditional expected excess returns have greatly affected out-of-sample performance of portfolios; thus, in this paper, the different targeting conditional expected returns, which are required by a mean-variance portfolio, are considered for q=(3%, 8%, 16%) annually. I also estimate for different portfolio sizes, n=(30, 50, 80, 100). The expected returns, , in equation (1), are computed as the average of realized returns over the estimation window of length, R.
The choice of structural shrinkage targeting matrices is also crucial. This paper takes four types of shrinkage targeting matrices for estimation: Shrinkage towards identity (I), Shrinkage towards market (M), Shrinkage towards constant correlation (C), and Shrinkage towards diagonal matrix (D), which are used in Ledoit and Wolf (2003) and Disatnik and Benninga (2007). I use S&P500 as market factor. The corresponding programming codes provided by these authors are available at http://www.ledoit.net/ research.htm.
6. Empirical Results
Using Ledoit and Wolf approach as benchmark, the cross validation performance is evaluated for out-of-sample portfolios in achieving: (i) larger standard deviation reduction; (ii) higher information ratio; and (iii) lower portfolio turnover. The turnover is defined as the total turnover of Grinold and Kahn (2000, Chapter 16) and DeMiguel et al. (2009). In general, the higher the turnover is, the less attractive the portfolio is to an active manager.
To measure the statistical significance of the difference between standard deviations and information ratios for two given portfolios, I apply bootstrapping methods. In particular, to compute p-values for the information ratios, I use the bootstrapping method proposed by Ledoit and Wolf (2008), while to test the hypothesis of the equality of two given portfolios’ variances, I employ the stationary bootstrap of Politis and Romano (1994), and then the resulting bootstrap p-values are generated by the method suggested by Ledoit and Wolf (2008, Remark 3.2). The programming code for the robust tests of Ledoit and Wolf (2008) is available at http://www.econ.uzh.ch/faculty/ wolf/ publications.html.
Table 1 reports the standard deviations of different out-of-sample portfolios and p-values in parenthesis. The portfolio standard deviations estimated from the shrinkage towards identity matrix is set as the benchmark, so that all other models are compared to it in terms of equality tests. The difference is statistically significant if the p-value is less than 5%. All other pair-wise p-values are also computed and used for discussion, while I do not report those due to the space limit. The averages of portfolio standard deviations of each criteria are also reported.
As seen in Table 1, the cross validation portfolios have always achieved the lower standard deviations than those of Ledoit and Wolf across different portfolio types, sizes, and conditional expected returns. On average, cross validation under the MinVariance performs the best for both the mean-variance and global minimum variance portfolios across the different required expected returns and portfolio sizes. For instance, despite that shrinkage towards market model (M) performs the best in Ledoit and Wolf portfolios, cross validation using MinVariance gets the lower standard deviations than the best performed portfolio in Ledoit and Wolf. Importantly, p-values also show that the outperformance of cross validation is statistically significant.
Besides the substantial reduction on average, it is more interestingly observed the consistent standard deviation reduction in individual targeting portfolios by cross validation. Specifically, the standard deviation estimated by MinVariance and MinDeviation through a shrinkage targeting matrix is generally smaller than that estimated by Ledoit and Wolf through the same targeting matrix. For instance, the standard deviation, estimated by the identity shrinkage targeting matrix for the global minimum variance portfolio with n=100, is 0.4015, smaller than that of the corresponding one from Ledoit and Wolf, 0.4339. Moreover, it is also smaller than the best estimation of Ledoit and Wolf by the shrinkage towards market, 0.4141. p-values show the difference is statistically significant. However, the reduction in the standard deviation cannot be found through maximizing information ratio when q=8% and 16%. This exception is, nevertheless, not surprising due to the trade-off of risk-return.
Table 2 presents the out-of-sample information ratios of various portfolios and p-values of the equality robust test. As shown, the information ratios estimated by cross validation are significantly higher than those of Ledoit and Wolf on average. The range of differences in information ratios is from 0.0072 to 0.0434 on average. In addition, cross validation under MinVariance and MinDeviation obtains the consistently higher information ratios than those of Ledoit and Wolf estimated by the same individual targeting matrix across various portfolios. Furthermore, the highest information ratios are always obtained by cross validation approach. The largest difference at individual targeting estimation is 0.0819 for the portfolio size, n=50, at q=8%.
As expected, in Table 2, cross validation by maximizing information ratios gives consistently higher information ratios than those of Ledoit and Wolf on average. Furthermore, its shrinkage towards diagonal matrix has obtained the highest information ratios among the competing methods including Ledoit and Wolf approach. However, although maximizing information ratios is set to maximizing out-of-sample portfolio information ratios, it does not universally achieve the highest information ratios through other individual targeting matrices. As a result, this evidence shows that the estimation by maximizing information ratios might have larger uncertainty than other methods.
From Table 3 it is observed that the turnovers of the cross validation portfolios are much lower than those of Ledoit and Wolf. On average, the difference is ranged from 0.15 to 1.136, while the largest difference is 2.63 between the portfolios with the size n=100 and q=1600 estimated by the individual targeting of identity matrix. In contrast to the performance in the standard deviation reduction and the information ratio improvement, cross validation by maximizing information ratios always has the lowest turnovers for the mean-variance portfolios among all competing methods. Table 3 also shows that shrinkage towards identity matrix from cross validation has obtained all the lowest turnovers for mean-variance portfolios, whereas shrinkage towards diagonal matrix of cross validation has always gotten the lowest turnovers for global minimum variance portfolios.
Additionally, it is interesting in observing the difference between the shrinkage intensities of cross validation and Ledoit and Wolf. Figure 1 plots the shrinkage intensities of the portfolio with the size n=50. The figures for other portfolios are similar. Figure 1 shows that the shrinkage intensities vary significantly across competing methods. The shrinkage intensities estimated by Ledoit and Wolf are lower than those by cross validation in general, except for the constant correlation targeting matrix. Interestingly, the shrinkage intensities estimated by cross validation are more volatile than those of Ledoit and Wolf. The shrinkage intensities of the constant correlation targeting matrix show an apparent structural break around July of 1997 from Ledoit and Wolf estimation.
To align loss functions of estimation and evaluation for out-of-sample portfolios, this paper suggests the non-parametric technique of leave-one-out cross validation for tuning the shrinkage intensity estimation. Compared to the shrinkage method of Ledoit and Wolf, the cross validation portfolios achieves both significant variance reduction and information ratio improvement across different expected returns and portfolio sizes. The criteria of minimizing out-of-sample variance and minimizing the deviation of the tracking portfolio from the expected returns consistently outperform Ledoit and Wolf approach. I also find that the shrinkage intensities estimated by cross validation are in general more volatile and higher than those of Ledoit and Wolf. Finally, the portfolios estimated by cross validation are more attractive to an active manager due to their lower portfolio turnovers.
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