Portfolio Performance regarding Estimation Risk

The following experiment aims to extend the EGP-IMG model for the existence of estimation risk with regard to the application of Stein estimation and Shrinkage approach in consequence. As it has been mentioned in Section 2:4, the major concern for well implementing the EGP-IMG model is to properly introduce the input parameters, that is, the …rst two moments of securities returns in (7) to (11), which are seldom known in practice and hard to estimate that the issue of estimation risk arises accordingly. The input parameters estimations in the preceding EGP-IMG model are basically derived by the two-factors model under the two-stage multi-group framework for it is presumed that securities returns are mainly a¤ected by the group indices and in turn by the market index returns and uncertain in‡ation rates. However, for decision-makers having personally (privately) prior belief in not even being in a weak-form e¢ cient capital market, or facing a small sample size T that multi-group regressions in the preceding EGP-IMG model may not serve well for yielding stable regression parameters in , , a, b, and c in (1) and (2) and then failing in input parameters estimations, they may consider directly introducing history sample moments in portfolio optimization for convenience. That means decision-makers at that time simply adopt sample means and sample covariances within a short period of time and treat them as if they were true parameters in the mean-variance framework. In such case, portfolio optimization based on the EGP-IMG model is to directly plug history sample means and covariances into the formula (7) to (11). But for the possibly growing estimation risk (errors) accompanied by the introduction of history sample estimators, Stein estimation and shrinkage approach respectively applied to sample means and sample covariances estimations are then recommended. It is for the reason that the critical determinant in yielding an outperformed portfolio in the MV framework is to obtain adequate and superior estimators of asset returns moments in the process of portfolio optimization. Here what de…ne “adequate and superior” estimators for true parameters of returns distributions are their precise and stable properties. Using history sample estimators incorporating Stein estimation and shrinkage approach is exact to try making e¤ort in balancing the trade-o¤ relation between stable and precise properties for the candidate estimators of true parameters.

Table 2:5 …rst displays the summary statistics of out-of-sample Sharpe ratios for port-folios constructed by the EGP-IMG model with respect to history sample estimators incorporating Stein estimation and shrinkage approach on means and covariances estima-tions. Table 2:6 further examines the di¤erence in out-of-sample Sharpe ratios for di¤erent portfolios relative to the benchmark portfolio. The benchmark portfolio here is simply referred to the HIS_HIS portfolio (strategy), which purely uses history sample means and history sample covariances as input parameters in the EGP-IMG model for optimal portfolio construction. We can observe that the adoption of pure history sample moments in the EGP-IMG model does not serve well for remarkable out-of-sample portfolio per-formance. It even yields the averagely negative Sharpe ratio in the short-sales allowed scenario (-0:01838). The results may be frustrating but not astonishing to decision-makers as it is well-known that sample estimators are much volatile rather than stable within a short period of time and the MV optimization is very sensitive to changes (errors) in the estimates of these inputs (Merton (1980); Best and Grauer (1991)[11]). Focusing on the pioneered works of Brown (1976)[18] and Bawa, Brown, and Klein (1979) that have

exam-ined the history pulg-in method is generally dominated by the Bayesian approach under a di¤use prior, we then propose the comparable portfolios, BS_HIS and JS_HIS, which implement Stein estimators on means in the EGP-IMG model for hopefully improved performance relative to the benchmark HIS_HIS portfolio in both short-sales allowed and long-only cases. Accordingly, HIS_LWCC and HIS_LWID are portfolios using the shrinkage approach on covarinaces estimation given the history sample means. Results in these two tables do approve that the adoption of Stein estimation is indeed helpful for getting better estimators of input parameters on means as portfolios BS_HIS and JS_HIS are able to exhibit obvious performance improvement relative to the HIS_HIS benchmark. However, portfolios HIS_LWCC and HIS_LWID fail to present remarkable performance improvement compared with the HIS_HIS portfolio. Introducing the shrink-age covariance matrix, the LWID matrix, even results in signi…cantly impaired Sharpe ratio value relative to the HIS_HIS benchmark in the short-sales allowed case. Though the other shrinkage matrix on covariances, the LWCC matrix, seems to slightly raise port-folio performance, it still does not signi…cantly outperform the benchmark. Performances on portfolios HIS_LWCC and HIS_LWID implicitly indicate that purely concerning es-timation risk on group covariances but ignoring eses-timation errors on securities means cannot signi…cantly contribute to better performance in terms of improving the history sample-based portfolio in the end.

Strategy Average Sharpe Ratio Standard Deviation Maximum Minimum Panel A: Short-Sales Allowed Case

HIS_HIS -0.01838 0.15754 0.38759 -0.39025

BS_HIS 0.06513 0.15876 0.51040 -0.41811

JS_HIS 0.06519 0.15878 0.51043 -0.41806

HIS_LWCC -0.01346 0.15884 0.38798 -0.39628

HIS_LWID -0.02296 0.15991 0.38841 -0.40053

BS_LWCC 0.06518 0.15788 0.50653 -0.41640

BS_LWID 0.06653 0.15869 0.51161 -0.41767

JS_LWCC 0.06524 0.15790 0.50656 -0.41635

JS_LWID 0.06659 0.15871 0.51163 -0.41763

Panel B: Long-Only Case

HIS_HIS 0.01528 0.21447 0.46228 -0.48926

BS_HIS 0.07233 0.16711 0.56327 -0.41463

JS_HIS 0.07237 0.16710 0.56330 -0.41458

HIS_LWCC 0.01829 0.21674 0.46312 -0.49294

HIS_LWID 0.01428 0.21474 0.46074 -0.48811

BS_LWCC 0.07238 0.16664 0.56069 -0.41266

BS_LWID 0.07255 0.16689 0.56226 -0.41434

JS_LWCC 0.07241 0.16663 0.56071 -0.41261

JS_LWID 0.07258 0.16688 0.56227 -0.41428

Table 2.5: Summary Statistics of Sharpe Ratios regarding Estimation Risk. Summary statistics on out-of-sample Sharpe ratios for di¤erent investment strategies based on weekly returns over the preceding 160 weeks. The …rst column gives the abbreviation used in referring to the strategy throughout the study. The term HIS-HIS is the benchmark portfolio using history sample means and history sample covariance matrix as the input parameters in the EGP-IMG model. BS-HIS is the portfolio introducing Bayes-Stein estimators on means and history sample covariance matrix as input parameters. The rest is deduced by analogy.

Short-Allowed Case Long-Only Case

BS_HIS 0.06513 0.08351* 0.07233 0.05705*

(2.0144) (1.8997)

JS_HIS 0.06519 0.08357* 0.07237 0.05709*

(2.0157) (1.9001)

HIS_LWCC -0.01346 0.00492 0.01829 0.00301

(0.9946) (0.7021)

HIS_LWID -0.02296 -0.00458* 0.01428 -0.00100

(1.9254) (0.6523)

BS_LWCC 0.06518 0.08356* 0.07238 0.05710*

(2.0339) (1.9032)

BS_LWID 0.06653 0.08491** 0.07255 0.05727*

(2.0391) (1.9037)

JS_LWCC 0.06524 0.08362* 0.07241 0.05713*

(2.0369) (1.9060)

JS_LWID 0.06659 0.08497** 0.07258 0.05730*

(2.0382) (1.9064)

Table 2.6: Performance Robustness Test regarding Estimation Risk. Statistic test on the di¤erence of weekly out-of-sample Sharpe ratios for pairwise portfolios. Statistic values from studentized circular block bootstrapping methodology developed by Ledoit and Wolf (2008) is displayed in the parentheses. The term HIS-HIS is the benchmark portfolio using history sample means and history sample covariance matrix as the input parameters in the EGP-IMG model. BS-HIS is the portfolio introducing Bayes-Stein estimators on means and history sample covariance matrix as input parameters. The rest is deduced by analogy. Symbols of ***, **, and* respectively refer to 0.01, 0.05, and 0.10 statistically signi…cant levels.

We …nally consider to simultaneously calibrate estimation risk on securities means and group covariacnes. Stein estimation applied to means estimation and shrinkage ap-proach to covariances measurement are incorporated in pairs in the EGP-IMG model that four portfolios, BS_LWCC, BS_LWID, JS_LWCC, and JS_LWID, are reformed accordingly. Results in tables clear indicate that performances on these four portfolios are actually able to dominate over the benchmark in both short-sales allowed and long-only cases. However, it deserves to be mentioned that their improved performances seem not to far outperform the BS_HIS and JS_HIS portfolios. The phenomenon implies that incorporating shrinkage approach on covariances estimation though somewhat contributes to slightly raise average Sharpe ratios of the history sample-based portfolio, the accom-panied performance improvement is not that impressed. Calibrating estimation errors on securities means by introducing Stein estimators instead dedicates more to yield a rela-tively outperformed portfolio. Our experiment results to the comparative importance of estimation risk (errors) on means and covariances imply that calibrating errors in means are much critical than calibrating errors in covariances for improving portfolio perfor-mance. The results are consistent with the works of Kallberg and Ziemba (1984)[46] and Chopra and Ziemba (1993)[22], which indicate the primary focus for portfolio

optimiza-tion in the MV framework should be on obtaining superior estimates of means. Errors in covariances on the contrary are rather minor in terms of their in‡uence on determining the constituents of optimal portfolio.

In addition, notice the phenomenon here that portfolio performances in the long-only case averagely dominate the counterparts without short-sales constraints. The experi-ment outcome is contrary to the preceding results in Table 2:3 and Table 2:4 using the two-factors model for input parameters estimations in the EGP-IMG model, which reveals performances are much better in the short-sales allowed case. The di¤erence in perfor-mance with regard to the impact of the short-sales constraints on the current experiment contrasts to that on the preceding analysis may mainly attribute to the adoption of his-tory sample moments by which estimation risk accompanies. For optimization process in the EGP-IMG model that is restricted to use history sample moments rather than the presumed two-factors model for true parameters estimations, estimation risk at that scenario is expected to be much considerable. Imposing the short-sales constraints at this scenario for portfolio optimization in the EGP-IMG model may help to give relatively moderate estimates of input parameters and thus contributes to less extreme investment weights and portfolio performance, which are bene…cial to the performance measurement in terms of Sharpe ratios. Since forcing the short-sales constraints in portfolio selecting can further assist in reducing unreasonably drastic shortfalls in portfolio weights, that is why Jagannathan and Ma (2003)[41] indicate that imposing the no-short-sales restric-tions theoretically should hurt, whereas empirical evidence often shows the contrary in the mean-variance framework. The …nding of Green and Holli…eld (1992)[39] also indicates that empirical results suggest imposing short-sales constraints can improve the e¢ ciency of optimal portfolios constructed using sample moments. In fact, it has long been recog-nized that portfolio optimization using sample means and sample covariances serves for poor out-of-sample performance. Jagannathan and Ma (2003) especially proclaim that the sample mean is an imprecise estimator of the population mean. The estimation error in the sample mean is usually so large. Our experiment therefore contributes to implic-itly give recommendation that if portfolio optimization is conducted based on the history sample moments and hence is forced to the relatively considerable estimation risk (errors), imposing the nonnegative portfolio weights is expected to be able to reduce estimation risk and bene…t ultimate portfolio performance.