A. Sample Estimates and Distributional Assumption

Given a sample of asset excess returns of size $ T $ , $ \left({\boldsymbol{r}}_1,\dots, {\boldsymbol{r}}_T\right) $ , let

(5) $$ {\hat{\boldsymbol{\mu}}}_N=\frac{1}{T}\sum \limits_{t=1}^T{\boldsymbol{r}}_t\hskip1em \mathrm{and}\hskip1em {\hat{\boldsymbol{\Sigma}}}_N=\frac{1}{T}\sum \limits_{t=1}^T\left({\boldsymbol{r}}_t-{\hat{\boldsymbol{\mu}}}_N\right){\left({\boldsymbol{r}}_t-{\hat{\boldsymbol{\mu}}}_N\right)}^{\prime } $$

be the sample estimates of $ {\boldsymbol{\mu}}_N $ and $ {\boldsymbol{\Sigma}}_N $ , and we require $ T>N $ so that $ {\hat{\boldsymbol{\Sigma}}}_N $ is almost surely invertible. Given a sample portfolio $ \hat{\boldsymbol{w}} $ estimated with $ {\hat{\boldsymbol{\mu}}}_N $ and $ {\hat{\boldsymbol{\Sigma}}}_N $ , we measure its performance with the EU pioneered by Kan and Zhou (Reference Kan and Zhou2007),

(6) $$ EU\left(\hat{\boldsymbol{w}}\right)=\unicode{x1D53C}\left[U\left(\hat{\boldsymbol{w}}\right)\right]=\unicode{x1D53C}\left[{\hat{\boldsymbol{w}}}^{\prime }{\boldsymbol{\mu}}_N-\frac{\gamma }{2}{\hat{\boldsymbol{w}}}^{\prime }{\boldsymbol{\Sigma}}_N\hat{\boldsymbol{w}}\right]. $$

We then define $ {N}^{\star } $ as the optimal portfolio size $ N $ maximizing the EU of $ \hat{\boldsymbol{w}} $ .

Kan and Zhou (Reference Kan and Zhou2007) and follow-up papers, such as Tu and Zhou (Reference Tu and Zhou2011), Kan et al. (Reference Kan, Wang and Zhou2021), Lassance et al. (Reference Lassance, Vanderveken and Vrins2024), Lassance, Martín-Utrera, and Simaan (Reference Lassance, Martín-Utrera and Simaan2024), and Yuan and Zhou (Reference Yuan and Zhou2024), evaluate (6) under the IID multivariate normal distributional assumption. Instead, we follow Kan and Lassance (Reference Kan and Lassance2025), who study the EU of various sample portfolios under the IID multivariate elliptical distribution. Like them, we use the stochastic representation of the elliptical distribution in El Karoui (Reference El Karoui2010), (Reference El Karoui2013).

We recover the normal distribution when $ \tau =1 $ and the $ t $ -distribution when $ \tau \sim \left(\nu -2\right)/{\chi}_{\nu}^2 $ , where $ {\chi}_{\nu}^2 $ is a chi-square distribution with $ \nu >2 $ degrees of freedom. We focus on the elliptical distribution because it is consistent with MV portfolios (Chamberlain (Reference Chamberlain1983), Schuhmacher, Kohrs, and Auer (Reference Schuhmacher, Kohrs and Auer2021)). Specifically, since the multivariate elliptical distribution is closed under linear transformations, all portfolios have the same higher moments and only differ in their mean return and variance. Therefore, for all increasing and concave utility functions, the optimal portfolio under expected utility is MV efficient. We also opt for the elliptical distribution because Kan and Lassance (Reference Kan and Lassance2025) show that accounting for fat tails is crucial in determining optimal portfolio combination rules.

B. Optimal Portfolio Size for Sample Portfolios

The first strategy we consider is the sample counterpart of the MV portfolio in (2), that is,

(7) $$ {\hat{\boldsymbol{w}}}^{\star }=\frac{1}{\gamma }{\hat{\boldsymbol{\Sigma}}}_N^{-1}{\hat{\boldsymbol{\mu}}}_N. $$

The SMV portfolio is not robust to parameter uncertainty because it is only optimal in sample. Therefore, we also consider the 2F of Kan and Zhou (Reference Kan and Zhou2007) that scales down the SMV portfolio to maximize its EU. This 2F is of the form

(8) $$ \hat{\boldsymbol{w}}\left(\alpha \right)=\alpha {\hat{\boldsymbol{w}}}^{\star }=\frac{\alpha }{\gamma }{\hat{\boldsymbol{\Sigma}}}_N^{-1}{\hat{\boldsymbol{\mu}}}_N, $$

where $ \alpha \in \mathrm{\mathbb{R}} $ is the combination coefficient. The SMV in (7) corresponds to $ {\hat{\boldsymbol{w}}}^{\star }=\overset{}{\hat{\boldsymbol{w}}}(\alpha =1) $ .

In the next proposition, we derive the EU of the 2F and, thus, also the SMV portfolio, the optimal combination coefficient $ {\alpha}^{\star } $ , and which EU it delivers, when asset returns are elliptically distributed. These results then allow us to determine the optimal portfolio size $ N $ . This proposition follows, with minor adjustments, from Kan and Lassance ((Reference Kan and Lassance2025), Propositions 7 and 8). In Appendix A.I, we follow the same approach for three-fund rules that extend the 2F by incorporating a third fully invested fund that is either the GMV or the EW portfolio. We refer to these as the GMV-three-fund rule (3FGMV) and the EW-three-fund rule (3FEW), respectively.

Proposition 2. Let $ T>N+4 $ , $ \boldsymbol{M}={\boldsymbol{I}}_T-{\mathbf{1}}_T{\mathbf{1}}_T^{\prime }/T $ , $ {\boldsymbol{Z}}_N $ be a $ T\times N $ matrix of independent standard normal variables, $ \boldsymbol{\Lambda} $ be a diagonal matrix of $ T $ independent copies of $ {\tau}^{1/2} $ , and $ \boldsymbol{\Lambda} \perp {\boldsymbol{Z}}_N $ . Then, under Assumption 2, the EU of the 2F $ \hat{\boldsymbol{w}}\left(\alpha \right) $ in (8) is

(9) $$ EU\left(\hat{\boldsymbol{w}}\left(\alpha \right)\right)=\frac{T}{2\gamma \left(T-N-2\right)}\left[\left(2{\alpha \kappa}_{N,1}-\frac{c_N{\alpha}^2{\kappa}_{N,2}T}{T-N-2}\right){\theta}_N^2-\frac{c_N{\alpha}^2{\kappa}_{N,3}N}{T-N-2}\right], $$

where $ {\theta}_N^2 $ is the maximum squared Sharpe ratio given in (4), $ {c}_N=\frac{\left(T-2\right)\left(T-N-2\right)}{\left(T-N-1\right)\left(T-N-4\right)} $ , and $ {\kappa}_{N,1} $ , $ {\kappa}_{N,2} $ , and $ {\kappa}_{N,3} $ , which we assume exist, are the functions of only $ N $ , $ T $ , and the distribution of $ \tau $ :

(10) $$ {\kappa}_{N,1}=\frac{T-N-2}{N}\unicode{x1D53C}\left[\mathrm{tr}\left({\left({\boldsymbol{Z}}_N^{\prime}\boldsymbol{\Lambda} \boldsymbol{M}\boldsymbol{\Lambda } {\boldsymbol{Z}}_N\right)}^{-1}\right)\right], $$

(11) $$ {\kappa}_{N,2}=\frac{{\left(T-N-2\right)}^2}{c_NN}\unicode{x1D53C}\left[\mathrm{tr}\left({\left({\boldsymbol{Z}}_N^{\prime}\boldsymbol{\Lambda} \boldsymbol{M}\boldsymbol{\Lambda } {\boldsymbol{Z}}_N\right)}^{-2}\right)\right], $$

(12) $$ {\kappa}_{N,3}=\frac{{\left(T-N-2\right)}^2}{c_N NT}\unicode{x1D53C}\left[{\mathbf{1}}_T^{\prime}\boldsymbol{\Lambda} {\boldsymbol{Z}}_N{\left({\boldsymbol{Z}}_N^{\prime}\boldsymbol{\Lambda} \boldsymbol{M}\boldsymbol{\Lambda } {\boldsymbol{Z}}_N\right)}^{-2}{\boldsymbol{Z}}_N^{\prime}\boldsymbol{\Lambda} {\mathbf{1}}_T\right]. $$

Moreover, the optimal combination coefficient $ {\alpha}^{\star } $ maximizing (9) and the resulting EU are

(13) $$ {\alpha}^{\star }=\frac{T-N-2}{c_NT}\left(\frac{\kappa_{N,1}{\theta}_N^2}{\kappa_{N,2}{\theta}_N^2+{\kappa}_{N,3}\frac{N}{T}}\right)\hskip0.66em \mathrm{and}\hskip0.66em EU\left(\hat{\boldsymbol{w}}\left({\alpha}^{\star}\right)\right)=\frac{1}{2\gamma {c}_N}\left(\frac{\kappa_{N,1}^2{\theta}_N^4}{\kappa_{N,2}{\theta}_N^2+{\kappa}_{N,3}\frac{N}{T}}\right). $$

Three comments are in order. First, Proposition 2 is valid for any $ {\boldsymbol{\Sigma}}_N $ , not only those of the form $ {\boldsymbol{\Sigma}}_N\left(\rho \right) $ . Second, $ {\kappa}_{N,1} $ , $ {\kappa}_{N,2} $ , and $ {\kappa}_{N,3} $ do not have a closed-form expression, but can be evaluated via Monte Carlo simulations given the distribution of $ \tau $ . We explain how we estimate them in Appendix A.II. Kan and Lassance (Reference Kan and Lassance2025) show that they increase with estimation risk $ N/T $ and tail heaviness. Third, under parameter uncertainty in $ {\boldsymbol{\mu}}_N $ and $ {\boldsymbol{\Sigma}}_N $ , a larger $ N $ may not always be favorable. Instead, the $ N $ maximizing (9) trades off between improving the population performance (i.e., increasing $ {\theta}_N^2 $ ) and limiting estimation risk (i.e., decreasing $ N/T $ ).

Next, we proceed as in Section II and relate the EU with $ N $ more explicitly by assuming that the covariance matrix $ {\boldsymbol{\Sigma}}_N $ complies with Assumption 1 (i.e., $ {\boldsymbol{\Sigma}}_N={\boldsymbol{\Sigma}}_N\left(\rho \right) $ ). We obtain this novel result by plugging the expression for $ {\theta}_N^2 $ under equicorrelation in (4) into equations (9) and (13).

Corollary 1. Under Assumptions 1 and 2, the EU of the SMV portfolio $ {\hat{\boldsymbol{w}}}^{\star } $ and the optimal 2F $ \hat{\boldsymbol{w}}\left({\alpha}^{\star}\right) $ are given by

(14) $$ EU\left({\hat{\boldsymbol{w}}}^{\star}\right)=\frac{NT}{2\gamma \left(T-N-2\right)}\left[\frac{\delta_N\left({\overline{\boldsymbol{\theta}}}_N\right)}{1-\rho}\left(2{\kappa}_{N,1}-\frac{c_N{\kappa}_{N,2}T}{T-N-2}\right)-\frac{c_N{\kappa}_{N,3}}{T-N-2}\right] $$

(15) $$ =: E{U}_{smv}\left(N,T,\rho, \tau, {\bar{\theta}}_N\right),\hskip0.33em \mathrm{and} $$

(16) $$ EU\left(\hat{\boldsymbol{w}}\left({\alpha}^{\star}\right)\right)=\frac{N}{2\gamma {c}_N\left(1-\rho \right)}\times \frac{\kappa_{N,1}^2{\delta}_N{\left({\overline{\boldsymbol{\theta}}}_N\right)}^2}{\kappa_{N,2}{\delta}_N\left({\overline{\boldsymbol{\theta}}}_N\right)+{\kappa}_{N,3}\frac{1-\rho }{T}}=: {EU}_{2f}\left(N,T,\rho, \tau, {\overline{\boldsymbol{\theta}}}_N\right). $$

$ {EU}_{smv} $ and $ {EU}_{2f} $ depend on several parameters.Footnote ⁹ First, the sample size $ T $ , directly but also via $ {c}_N $ and $ \left({\kappa}_{N,1},{\kappa}_{N,2},{\kappa}_{N,3}\right) $ . Second, the portfolio size $ N $ , directly but also via $ {c}_N $ , the function $ {\delta}_N $ , and $ \left({\kappa}_{N,1},{\kappa}_{N,2},{\kappa}_{N,3}\right) $ . Third, which $ N $ assets are selected via $ {\overline{\boldsymbol{\theta}}}_N=\left({\overline{\theta}}_{N,1},{\overline{\theta}}_{N,2}\right) $ .

Now, to maximize $ {EU}_{smv}\left(N,T,\rho, \tau, {\overline{\boldsymbol{\theta}}}_N\right) $ or $ {EU}_{2f}\left(N,T,\rho, \tau, {\overline{\boldsymbol{\theta}}}_N\right) $ with respect to $ N $ , one would also need to optimize the selection of assets because $ {\overline{\boldsymbol{\theta}}}_N $ depends on $ N $ via the chosen subset of assets. This is a difficult optimization problem because the number of possible subsets of $ N $ assets with $ N $ going from 1 to $ M $ grows very quickly with $ M $ .Footnote ¹⁰ Moreover, this would introduce potentially severe estimation risk and time instability in the selection of assets, which is undesirable. This is why, to disentangle the determination of the optimal $ N $ from the asset selection, we replace $ {\overline{\boldsymbol{\theta}}}_N $ by its counterpart for the whole investment universe (i.e., $ {\overline{\boldsymbol{\theta}}}_M $ ). This approximation may not be accurate when $ N $ is small relative to $ M $ .Footnote ¹¹ In this case, however, the estimated $ {\overline{\boldsymbol{\theta}}}_N $ will be particularly noisy, and thus estimating it from $ {\overline{\boldsymbol{\theta}}}_M $ can help too. All in all, the way we determine the optimal portfolio size $ N $ for the SMV portfolio and the optimal 2F becomesFootnote ¹²

(17) $$ {N}_{smv}^{\star }=\underset{N\in \left\{1,\dots, \min \left(M,T-5\right)\right\}}{\arg \max }{EU}_{smv}\left(N,T,\rho, \tau, {\overline{\boldsymbol{\theta}}}_M\right),\mathrm{and} $$

(18) $$ {N}_{2f}^{\star }=\underset{N\in \left\{1,\dots, \min \left(M,T-5\right)\right\}}{\arg \max }{EU}_{2f}\left(N,T,\rho, \tau, {\overline{\boldsymbol{\theta}}}_M\right). $$

We now illustrate $ {N}_{smv}^{\star } $ and $ {N}_{2f}^{\star } $ . Using the full sample of the 96S-BM data set introduced in Section II, we obtain $ {\hat{\boldsymbol{\mu}}}_{96} $ and $ {\hat{\boldsymbol{\Sigma}}}_{96} $ using (5). Then, we set the population mean $ {\mu}_M={\hat{\mu}}_{96} $ and the population covariance matrix $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\left(\rho \right)={\hat{\boldsymbol{D}}}_{96}{\boldsymbol{P}}_{96}\left(\rho \right){\hat{\boldsymbol{D}}}_{96} $ with $ {\hat{\boldsymbol{D}}}_{96}=\operatorname{diag}{\left({\hat{\boldsymbol{\Sigma}}}_{96}\right)}^{1/2} $ , which yields $ {\overline{\boldsymbol{\theta}}}_{96}=\left(\mathrm{0.125,0.0169}\right) $ , and we take $ \rho \in \left\{\mathrm{0.2,0.5,0.8}\right\} $ . We consider either the normal distribution (i.e., $ \tau =1 $ and $ {\kappa}_{N,1}={\kappa}_{N,2}={\kappa}_{N,3}=1 $ ) or the $ t $ -distribution (i.e., $ \tau \sim \left(\nu -2\right)/{\chi}_{\nu}^2 $ , with $ \nu =6) $ . Figure 2 depicts $ {N}_{smv}^{\star } $ and $ {N}_{2f}^{\star } $ as a function of $ T $ . We make several observations. First, $ {N}_{smv}^{\star } $ and $ {N}_{2f}^{\star } $ increase with $ T $ because, as $ T $ increases, the estimated portfolios become better estimates of the true MV portfolio for which the optimal $ N=M $ .

FIGURE 2 Optimal Portfolio Size for the Sample Mean–Variance (SMV) portfolio and the Optimal Two-Fund Rule (2F)

Figure 2 depicts the optimal portfolio size for the SMV portfolio, $ {N}_{smv}^{\star } $ in (17) (Graph A), and for the optimal 2F, $ {N}_{2f}^{\star } $ in (18) (Graph B), as a function of the sample size $ T $ . Each line represents a different choice of the correlation, $ \rho =\left\{\mathrm{0.2,0.5,0.8}\right\} $ , and the degrees of freedom of the $ t $ -distribution, $ \nu =\left\{6,\infty \right\} $ . We calibrate $ {\overline{\boldsymbol{\theta}}}_M=\left(\mathrm{0.125,0.0169}\right) $ to a data set of $ M=96 $ portfolios sorted on size and book-to-market spanning July 1963 to August 2023.

Second, $ {N}_{smv}^{\star } $ is very small compared with $ T $ and $ M $ . For instance, with $ T=240 $ and $ \rho =0.5 $ , $ {N}_{smv}^{\star }=3 $ both for the normal and $ t $ -distributions. For a larger $ \rho =0.8 $ , $ {N}_{smv}^{\star } $ gets larger but is still small. For example, when $ \rho =0.8 $ and $ T=120 $ , 180, and 240, we have $ {N}_{smv}^{\star }=1 $ , $ 3 $ , and 10 for the normal distribution, and $ {N}_{smv}^{\star }=1 $ , $ 2 $ , and 7 for the $ t $ -distribution, respectively.

Third, $ {N}_{smv}^{\star } $ and $ {N}_{2f}^{\star } $ tend to increase with $ \rho $ . This is because, as we show in Section OA.2 of the Supplementary Material, the maximum utility without parameter uncertainty, $ U\left({\boldsymbol{w}}^{\star}\right) $ , is a convex function of $ \rho $ and, thus, is increasing in $ \rho $ for $ \rho $ large enough. Therefore, given that $ U\left({\boldsymbol{w}}^{\star}\right) $ is an increasing function of $ N $ , a larger $ \rho $ tends to increase the optimal $ N $ .

Fourth, $ {N}_{2f}^{\star } $ is higher than $ {N}_{smv}^{\star } $ , below $ T/2 $ , equal to $ M $ if $ T/2 $ is enough above $ M $ , and gets closer to $ T/2 $ as $ \rho $ increases. These results can be explained based on (16) and (18). Specifically, we can show that in the normal case, that is, $ {\kappa}_{N,1}={\kappa}_{N,2}={\kappa}_{N,3}=1 $ , $ {N}_{2f}^{\star } $ is below $ T/2 $ (and equal to $ M $ if $ M $ is sufficiently below $ T/2 $ ), and close to $ T/2 $ when $ \rho $ is close to 0 or 1. To see this, consider the $ N $ maximizing the first term of $ {EU}_{2f} $ (i.e., $ N/{c}_N $ ). We have

(19) $$ \frac{N}{c_N}=\frac{N\left(T-N-1\right)\left(T-N-4\right)}{\left(T-2\right)\left(T-N-2\right)}\approx \frac{N\left(T-N-4\right)}{\left(T-2\right)}, $$

and the latter is maximized by

(20) $$ \underset{N\in \left\{1,\dots, \min \left(M,T-5\right)\right\}}{\arg \max}\frac{N\left(T-N-4\right)}{\left(T-2\right)}=\min \left(\left[T/2-2\right],M\right). $$

Moreover, when $ {\kappa}_{N,1}={\kappa}_{N,2}={\kappa}_{N,3}=1 $ , the second term of $ {EU}_{2f} $ in (16) decreases with $ N $ and is maximized by $ N=1 $ , which moves $ {N}_{2f}^{\star } $ below (20). However, the sensitivity of this second term to $ N $ depends on how sensitive $ N\rho /\left(1-\rho + N\rho \right) $ is to $ N $ , and it is less so as $ \rho $ approaches 0 or 1. In equity data where $ \rho $ is typically large, we thus expect $ {N}_{2f}^{\star } $ to stay close to (20). Finally, in the elliptical case, $ {\kappa}_{N,1} $ , $ {\kappa}_{N,2} $ , and $ {\kappa}_{N,3} $ also depend on $ N $ , but Figure 2 and later simulations suggest that $ {N}_{2f}^{\star } $ still remains close to that under normality.

Finally, fat tails positively impact $ {N}_{2f}^{\star } $ , whereas we observe the opposite for $ {N}_{smv}^{\star } $ . This can be explained because the 2F optimally scales down the SMV portfolio according to tail heaviness via $ {\kappa}_{N,1} $ , $ {\kappa}_{N,2} $ , and $ {\kappa}_{N,3} $ . Therefore, as shown by Kan and Lassance ((Reference Kan and Lassance2025), Proposition 5), the 2F often performs better when asset returns are elliptically instead of normally distributed, and thus we find a larger $ {N}_{2f}^{\star } $ under elliptical returns.

A. Estimated Optimal Portfolio Size

We now analyze the estimated optimal portfolio size for the SMV portfolio $ \left({\hat{N}}_{smv}^{\star}\right) $ , the 2F $ \left({\hat{N}}_{2f}^{\star}\right) $ , the 3FGMV $ \left({\hat{N}}_{3f,g}^{\star}\right) $ , and the 3FEW $ \left({\hat{N}}_{3f, ew}^{\star}\right) $ . These are the estimated counterparts of $ {N}_{smv}^{\star } $ in (17), $ {N}_{2f}^{\star } $ in (18), $ {N}_{3f,g}^{\star } $ in (A.10), and $ {N}_{3f, ew}^{\star } $ in (A.19) following the estimation methodology in Appendix A.II.

We simulate $ K=\mathrm{10,000} $ times $ T=120 $ returns and estimate the optimal $ N $ in each simulation. Figure 3 depicts the boxplots of $ {\hat{N}}^{\star } $ for the different strategies and choices of ( $ \nu $ , $ {\boldsymbol{\Sigma}}_M $ ). We make several observations. First, when $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right) $ , $ {\hat{N}}^{\star } $ is close to the oracle $ {N}^{\star } $ , highlighting the quality of our estimation procedure. Second, $ {\hat{N}}^{\star } $ is similar under $ {\hat{\boldsymbol{\Sigma}}}_{96} $ and $ {\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right) $ , meaning that drawing returns from a distribution violating Assumption 1 does not materially affect the estimation of $ N $ . Third, $ {\hat{N}}_{smv}^{\star } $ is very small, in line with Figure 2. Fourth, $ {\hat{N}}^{\star } $ is similar across the two-fund and three-fund rules and is typically slightly below $ T/2 $ , which is because $ \overline{\rho} $ is rather large (see Section III.B). Finally, the lower the $ \nu $ _, the higher the $ {\hat{N}}^{\star } $ in the two-fund and three-fund rules, in line with Figure 2.

FIGURE 3 Estimated Optimal Portfolio Size in Simulated Data

Figure 3 depicts the boxplots of the estimated optimal portfolio size $ N $ for the sample mean–variance portfolio $ \left({\hat{N}}_{smv}^{\star}\right) $ , the two-fund rule $ \Big({\hat{N}}_{2f}^{\star } $ ), the GMV-three-fund rule $ \left({\hat{N}}_{3f,g}^{\star}\right) $ , and the EW-three-fund rule $ \left({\hat{N}}_{3f, ew}^{\star}\right) $ . These are the estimated counterparts of $ {N}_{smv}^{\star } $ in (17), $ {N}_{2f}^{\star } $ in (18), $ {N}_{3f,g}^{\star } $ in (A.10), and $ {N}_{3f, ew}^{\star } $ in (A.19) following the estimation methodology in Appendix A.II. The boxplots are obtained by simulating $ \mathrm{10,000} $ times $ T=120 $ $ t $ -distributed returns. Using a data set of $ M=96 $ portfolios sorted on size and book-to-market spanning July 1963 to August 2023, we set $ {\boldsymbol{\mu}}_M={\hat{\boldsymbol{\mu}}}_{96} $ and $ {\boldsymbol{\Sigma}}_M\in \left\{{\hat{\boldsymbol{\Sigma}}}_{96},{\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right)\right\} $ , where $ {\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right) $ is an equicorrelation covariance matrix and $ \overline{\rho}=0.74 $ is the average of all correlations in $ {\hat{\boldsymbol{\Sigma}}}_{96} $ . We consider $ \nu \in \left\{6,\infty \right\} $ degrees of freedom. Each boxplot corresponds to a different choice of $ \left(\nu, {\boldsymbol{\Sigma}}_M\right) $ . We depict with crosses the oracle value of the optimal $ N $ that is known under Assumption 1 (i.e., when $ {\boldsymbol{\Sigma}}_M $ is of the form $ {\boldsymbol{\Sigma}}_M\left(\rho \right) $ ).

Next, in Figure 4, we let $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\left(\rho \right) $ and depict the oracle $ {N}^{\star } $ and boxplots of $ {\hat{N}}^{\star } $ for $ \rho $ varying between 0.1 and 0.9. We set $ \nu =6 $ ; the results are similar for $ \nu =\infty $ . We observe the following: First, the oracle $ {N}^{\star } $ is slightly below $ T/2 $ when $ \rho $ is close enough to 1, in line with Figure 2, where $ \rho =\overline{\rho}=0.74 $ . In this case of high correlation, typical for equity data, $ {\hat{N}}^{\star } $ is remarkably robust to parameter variability as the boxplots are thin. Second, for the SMV, 2F, and 3FGMV portfolios, the oracle $ {N}^{\star } $ first decreases with $ \rho $ and then reincreases, attaining its maximum as $ \rho $ approaches 1. This is because we want to invest in more assets when the maximum Sharpe ratio, $ {\theta}_N $ in (4), is larger, which also first decreases then reincreases with $ \rho $ (see Section OA.2 of the Supplementary Material). For $ \rho $ away from 0 and 1, the boxplots widen as $ {\hat{N}}^{\star } $ becomes more sensitive to the parameters. Third, $ \rho $ has a different effect for the 3FEW portfolio: $ {N}^{\star } $ decreases with $ \rho $ . This can be explained because 3FEW combines SMV with EW, which is not subject to estimation risk. Thus, when $ \rho $ decreases and $ {\theta}_N $ decreases too, 3FEW mostly invests in the EW portfolio and it is optimal to have a large $ N $ even under estimation risk. Finally, $ {\hat{N}}^{\star } $ is overall close to $ {N}^{\star } $ on average.

FIGURE 4 Impact of Correlation $ \rho $ on Optimal Portfolio Size in Simulated Data

Figure 4 depicts the boxplots of the estimated optimal portfolio size $ N $ for the sample mean–variance portfolio (Graph A, $ {\hat{N}}_{smv}^{\star } $ ), the two-fund rule (Graph B, $ {\hat{N}}_{2f}^{\star } $ ), the GMV-three-fund rule (Graph C, $ {\hat{N}}_{3f,g}^{\star } $ ), and the EW-three-fund rule (Graph D, $ {\hat{N}}_{3f,ew}^{\star } $ ). These are the estimated counterparts of $ {N}_{smv}^{\star } $ in (17), $ {N}_{2f}^{\star } $ in (18), $ {N}_{3f,g}^{\star } $ in (A.10), and $ {N}_{3f, ew}^{\star } $ in (A.19) following the estimation methodology in Appendix A.II. The boxplots are obtained by simulating $ \mathrm{10,000} $ times $ T=120 $ $ t $ -distributed returns. Using a data set of $ M=96 $ portfolios sorted on size and book-to-market spanning July 1963 to August 2023, we set $ {\boldsymbol{\mu}}_M={\hat{\boldsymbol{\mu}}}_{96} $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\left(\rho \right) $ , where $ {\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right) $ is an equicorrelation covariance matrix and $ \rho $ varies between 0.1 and 0.9 with a step size of 0.1. We consider $ \nu =6 $ degrees of freedom. We depict with dotted lines and crosses the oracle value $ {N}^{\star } $ of the optimal $ N $ that is known under Assumption 1 (i.e., when $ {\boldsymbol{\Sigma}}_M $ is of the form $ {\boldsymbol{\Sigma}}_M\left(\rho \right) $ ).

B. Performance of Size-Optimized Portfolios

We now evaluate the performance of the size-optimized SMV, two-fund, and three-fund rules using $ N={\hat{N}}^{\star } $ . For each choice of ( $ \nu $ , $ {\boldsymbol{\Sigma}}_M\Big) $ , where $ \nu \in \left\{6,\infty \right\} $ and $ {\boldsymbol{\Sigma}}_M\in \left\{{\hat{\boldsymbol{\Sigma}}}_{96},{\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right)\right\} $ , we draw $ K=\mathrm{10,000} $ times $ L=300 $ $ t $ -distributed returns, which we use in a rolling-window exercise. Specifically, in each simulation $ k $ , given the sample size $ T=120 $ and a chosen $ N $ , we randomly select in each rolling window a subset of $ N $ assets,Footnote ¹³ estimate the portfolio on these $ N $ assets (see our estimation methodology in Appendix A.II), and evaluate the out-of-sample portfolio return $ {r}_{t,k} $ on the next month. We also implement the EW portfolio, $ {\boldsymbol{w}}_{ew}={\mathbf{1}}_N/N $ , on the same $ N $ assets for comparison. We then roll the window by 1 month and proceed similarly until we reach the end of the sample of $ L $ returns. This gives us, for any $ N $ , a time series of out-of-sample portfolio returns $ {r}_{t,k} $ , $ t=1,\dots, L-T $ and $ k=1,\dots, K $ , and we compute the EU as

(21) $$ EU=\frac{1}{K}\sum \limits_{k=1}^K\left({\hat{\mu}}_k-\frac{\gamma }{2}{\hat{\sigma}}_k^2\right)\hskip1em \mathrm{with}\hskip1em {\hat{\mu}}_k=\frac{1}{L-T}\sum \limits_{t=1}^{L-T}{r}_{t,k},{\hat{\sigma}}_k^2=\frac{1}{L-T}\sum \limits_{t=1}^{L-T}{\left({r}_{t,k}-{\hat{\mu}}_k\right)}^2. $$

In Figure 5, we vary $ N $ from 5 to $ M=96 $ and depict the results for $ \nu =6 $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96} $ , which is the most relevant case. In Section OA.6.1 of the Supplementary Material, we set $ \nu =\infty $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96} $ , and in Section OA.6.2, $ \nu \in \left\{6,\infty \right\} $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\left(\overline{\rho}\right) $ . Graphs A–D in Figure 5 consider the SMV, 2F, 3FGMV, and 3FEW portfolios. Each graph depicts the EU as a function of $ N $ with a solid line using $ \gamma =1 $ and also of the EW portfolio with a dash-dotted red line. In addition, we depict with an horizontal dashed line the EU obtained by using the estimated optimal portfolio size, $ N={\hat{N}}_{t,k}^{\star } $ , which varies across rolling windows and simulations.Footnote ¹⁴

FIGURE 5 Expected Out-of-Sample Utility in Simulated Data $ \left(\nu =6,{\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\right) $

Figure 5 depicts the expected out-of-sample utility (EU) of the sample mean–variance portfolio (SMV; Graph A), the two-fund rule (2F; Graph B), the GMV-three-fund rule (3FGMV; Graph C), and the EW-three-fund rule (3FEW; Graph B) as a function of the portfolio size $ N $ . We simulate $ \mathrm{10,000} $ samples of $ t $ -distributed returns with $ \nu =6 $ degrees of freedom. For each $ N $ , we conduct a rolling window exercise as described in Section IV.B, and in each rolling window, we randomly select the $ N $ assets. Using a data set of $ M=96 $ portfolios sorted on size and book-to-market spanning July 1963 to August 2023, we set $ {\boldsymbol{\mu}}_M={\hat{\boldsymbol{\mu}}}_{96} $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96} $ . In each graph, the solid blue line depicts the EU in (21), the shaded gray area depicts the one-sigma interval around the EU across simulations, and the dashed horizontal blue line depicts the EU obtained when using the estimated optimal $ N $ (i.e., $ {\hat{N}}_{smv}^{\star } $ for the SMV portfolio, $ {\hat{N}}_{2f}^{\star } $ for 2F, $ {\hat{N}}_{3f,g}^{\star } $ for 3FGMV, and $ {\hat{N}}_{3f, ew}^{\star } $ for 3FEW). The dash-dotted red line depicts the EU of the equal-weighted portfolio. The dotted horizontal gray line depicts the zero EU level. The risk-aversion coefficient is $ \gamma =1 $ .

We make several observations. First, $ {\hat{N}}^{\star } $ delivers an EU close to the maximum, even though equicorrelation does not hold, there is estimation error in $ {\hat{N}}^{\star } $ , and the asset selection is random.Footnote ¹⁵ Second, for 2F, 3FGMV, and 3FEW, the EU is maximized for $ N $ slightly below $ T/2 $ , in line with Section III.B and the average correlation being large ( $ \overline{\rho}=0.74 $ ). In contrast, the optimal $ N $ is very small for SMV, as shown in Figure 2. Third, for 2F, 3FGMV, and 3FEW, $ {\hat{N}}^{\star } $ delivers an EU substantially larger than that when $ N $ is too small or too close to $ M $ , highlighting the value of choosing the right portfolio size and the cost of blindly investing in all assets. This EU obtained with $ {\hat{N}}^{\star } $ substantially outperforms the naive EW portfolio, which is a challenging benchmark for $ M $ close to $ T $ .

C. Comparison with Benchmarks

We now consider benchmark portfolio strategies that we also consider in the empirical analysis of Section V. First, we consider two natural ways of reducing portfolio size, which are soft- and hard-thresholding versions of the SMV portfolio, called SMV-ST and SMV-HT.

SMV-ST solves the MV portfolio problem subject to an $ {L}_1 $ -norm constraint:

(22) $$ {\hat{\boldsymbol{w}}}_{SMV- ST}=\underset{\boldsymbol{w}\in {\mathrm{\mathbb{R}}}^M}{\arg \max}\;{\boldsymbol{w}}^{\prime }{\hat{\boldsymbol{\mu}}}_M-\frac{\gamma }{2}{\boldsymbol{w}}^{\prime }{\hat{\boldsymbol{\Sigma}}}_M\boldsymbol{w}\hskip1em \mathrm{subject}\ \mathrm{to}\hskip1em {\left\Vert \boldsymbol{w}\right\Vert}_1=\sum \limits_{i=1}^M\mid {w}_i\mid \le \delta . $$

The $ {L}_1 $ -norm constraint performs asset selection and limits short-selling, which helps improve out-of-sample performance (DeMiguel et al. (Reference DeMiguel, Garlappi, Nogales and Uppal2009a)). We calibrate the threshold $ \delta $ using cross-validation with out-of-sample utility as a decision criterion. Specifically, we search the optimal $ \delta $ in an interval of 1,000 equally spaced values ranging from 0 to $ {\delta}_{\mathrm{max}} $ , defined as the norm of the unconstrained solution to (22) (i.e., $ {\delta}_{\mathrm{max}}={\left\Vert \frac{1}{\gamma }{\hat{\boldsymbol{\Sigma}}}_M^{-1}{\hat{\boldsymbol{\mu}}}_M\right\Vert}_1 $ ).

SMV-HT selects the portfolio size by keeping only the assets whose absolute SMV portfolio weight is above a threshold $ \overline{w} $ . In that case, the value of $ N $ is determined by $ \overline{w} $ as

(23) $$ {\hat{N}}_{SMV- HT}=\sum \limits_{i=1}^M{\unicode{x1D7D9}}_{\left\{|{\hat{w}}_i^{\star }|\ge \overline{w}\right\}},\hskip1em \mathrm{where}\hskip1em {\hat{\boldsymbol{w}}}^{\star }=\frac{1}{\gamma }{\hat{\boldsymbol{\Sigma}}}_M^{-1}{\hat{\boldsymbol{\mu}}}_M. $$

Once the $ {\hat{N}}_{SMV- HT} $ assets are selected, we recompute the SMV portfolio on these assets, which delivers better performance than keeping the original weights. We select the threshold $ \overline{w} $ via cross-validation with out-of-sample utility as a decision criterion. Specifically, we search $ \overline{w} $ in an interval of 10 equally spaced values ranging from 0 (all assets selected) to $ {\overline{w}}_{\mathrm{max}} $ (no asset selected), where $ {\overline{w}}_{\mathrm{max}}=\max \mid \frac{1}{\gamma }{\hat{\boldsymbol{\Sigma}}}_M^{-1}{\hat{\boldsymbol{\mu}}}_M\mid $ is the largest absolute weight of the full SMV portfolio.Footnote ¹⁶

In Figure 6, we depict the EU of SMV-ST and SMV-HT as a function of the number of assets on which they are estimated using the same setup as in Figure 5, with $ \nu =\infty $ .Footnote ¹⁷ We also depict boxplots of the estimated optimal $ N $ under SMV-ST and SMV-HT. We make several observations. First, SMV-ST and SMV-HT substantially improve upon the plain SMV portfolio in Figure 5. Second, SMV-ST and SMV-HT deliver an EU that is consistently negative. Therefore, they are less effective approaches to reducing portfolio size than our size-optimized two-fund and three-fund rules shown in Figure 5. Third, SMV-ST and SMV-HT set a small optimal $ N $ , in line with the small EU-optimal portfolio size for the SMV portfolio that we obtain theoretically (see Figure 2).

FIGURE 6 Expected Out-of-Sample Utility and Estimated Portfolio Size of Soft- and Hard-Thresholding of the Sample Mean–Variance Portfolio in Simulated Data $ \left(\nu =6,{\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\right) $

Figure 6 depicts the expected out-of-sample utility (EU) and the estimated optimal portfolio size of the soft- and hard-thresholding versions of the sample mean–variance (SMV) portfolio, SMV-ST and SMV-HT, described in Section IV.C. We depict these as a function of the number of assets $ N $ on which SMV-ST and SMV-HT are estimated. We simulate $ \mathrm{10,000} $ samples of $ t $ -distributed returns with $ \nu =6 $ degrees of freedom. For each $ N $ , we conduct a rolling window exercise as described in Section IV.B, and in each rolling window, we randomly select the $ N $ assets. Using a data set of $ M=96 $ portfolios sorted on size and book-to-market spanning July 1963 to August 2023, we set $ {\boldsymbol{\mu}}_M={\hat{\boldsymbol{\mu}}}_{96} $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96} $ . In Graphs A and C, the solid blue lines depict the EU of SMV-ST or SMV-HT in (21), the shaded gray areas depict the one-sigma interval around the EU across simulations, the dash-dotted red lines depict the EU of the equal-weighted portfolio, and the dotted horizontal gray lines depict the zero EU level. The risk-aversion coefficient is $ \gamma =1 $ . In Graphs B and D, the blue crosses depict the average estimated optimal portfolio size.

The final benchmark imposes a small-dimensional factor model in the estimation. Specifically, as detailed in Section OA.4 of the Supplementary Material, we construct an F+A strategy, where the factor part is estimated via PCA, the alpha part via the arbitrage portfolio methodology of Da et al. (Reference Da, Nagel and Xiu2024), and the factor and alpha portfolios are combined to maximize expected utility. This is a relevant benchmark for two reasons. First, although it does not reduce the portfolio size, it still reduces the number of parameters to estimate due to the factor model. Second, Da et al. ((Reference Da, Nagel and Xiu2024), Theorem 4) demonstrate that in the limit as $ N $ and $ T $ get large, and under specific assumptions, their arbitrage portfolio attains the maximum achievable Sharpe ratio over all portfolios having zero factor exposure. Therefore, the F+A strategy might work well even for a large $ N $ .

In Figure 7, we consider the same setup as in Figures 5 and 6 and depict the EU delivered by the F+A strategy as a function of $ N $ . In Graph A, the sample size $ T=120 $ . In Graph B, $ T=60+2N $ increases with $ N $ to capture the asymptotics in Da et al. (Reference Da, Nagel and Xiu2024). For comparison, we also depict the EU of the 2F; we find a similar conclusion for three-fund rules. We observe that when $ T $ is fixed, the EU of F+A decreases with $ N $ and substantially underperforms the 2F and EW portfolios. However, when $ T $ increases with $ N $ , the EU of F+A tends to improve with $ N $ and is maximized at $ N=M $ , where it slightly outperforms the EW portfolio. Nonetheless, the EU of the 2F strategy also improves with $ N $ in that case and outperforms F+A for all $ N $ . Overall, these results suggest that in our simulation setting, the 2F strategy delivers a better out-of-sample performance under estimation risk than the F+A strategy constructed from Da et al. (Reference Da, Nagel and Xiu2024).

FIGURE 7 Expected Out-of-Sample Utility of Factor-Plus-Alpha Strategy in Simulated Data $ \left(\nu =6,{\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96}\right) $

Figure 7 depicts the expected out-of-sample utility (EU) of the factor-plus-alpha (F+A) strategy, described in Section OA.4 of the Supplementary Material, as well as the two-fund rule, as a function of the portfolio size $ N $ . We simulate $ \mathrm{10,000} $ samples of $ t $ -distributed returns with $ \nu =6 $ degrees of freedom. For each $ N $ , we conduct a rolling window exercise as described in Section IV.B, and in each rolling window, we randomly select the $ N $ assets out of the $ M $ available ones. Using a data set of $ M=96 $ portfolios sorted on size and book-to-market spanning July 1963 to August 2023, we set $ {\boldsymbol{\mu}}_M={\hat{\boldsymbol{\mu}}}_{96} $ and $ {\boldsymbol{\Sigma}}_M={\hat{\boldsymbol{\Sigma}}}_{96} $ . In each graph, the solid blue and dashed green lines depict the EU in (21) of F+A and 2F, respectively, the shaded gray area depicts the one-sigma interval around the EU of F+A across all simulations, the dash-dotted red line depicts the EU of the equal-weighted portfolio, and the dotted horizontal gray line depicts the zero EU level. In Graph A, the sample size $ T=120 $ is fixed. In Graph B, the sample size increases with $ N $ as $ T=60+2N $ . The risk-aversion coefficient is $ \gamma =1 $ .

A. Data Sets

Our empirical analysis is based on six data sets of characteristic and industry portfolios and one data set of 100 individual stocks, which we list in Table 1 with the time period considered for each. Given that our objective is to reduce the portfolio size, the full investment universe must not be too small in the first place, and thus we consider data sets with $ M $ around 100.

TABLE 1 List of Data Sets Considered in the Empirical Analysis

The first data set already introduced in Section II, 96S-BM, is composed of decile portfolios sorted on size and book-to-market. The second data set, 108CHA, is composed of the long and short legs of the 54 characteristics considered in Lassance and Martín-Utrera (Reference Lassance and Martín-Utrera2025), collected from the authors. The third data set, 100S-OP, is composed of decile portfolios sorted on size and operating profitability. The fourth data set, 94IN-NV, is composed of 48 industry portfolios and the long and short legs obtained from the 23 characteristics in Novy-Marx and Velikov (Reference Novy-Marx and Velikov2015), available on Robert Novy-Marx’s website. The fifth data set, 107IN-CHA, is composed of 47 industry portfolios,Footnote ¹⁸ quintile portfolios sorted on size and book-to-market and on operating profitability and investment, and decile portfolios sorted on momentum. The sixth data set, 98IN-CHA-NV, is composed of 47 industry portfolios, quintile portfolios sorted on size and book-to-market, decile portfolios sorted on momentum, and the long and short legs obtained from the eight low-turnover characteristics in Novy-Marx and Velikov (Reference Novy-Marx and Velikov2015). For the seventh and final data set, 100STO, we follow Lassance et al. (Reference Lassance, Vanderveken and Vrins2024) and collect adjusted returns from CRSP for the 235 stocks traded on the three major U.S. stock exchanges that traded between 1998 and 2022. We consider 100 data sets of size $ M=100 $ , randomly drawn from the total pool of 235 stocks, and report the portfolio performance after merging the out-of-sample portfolio returns across the 100 data sets.

In Section OA.7.1 of the Supplementary Material, we look at how Assumption 1 (equicorrelated returns) and Assumption 2 (elliptical returns) transpire in the seven data sets. First, the equicorrelation assumption is reasonable for the 96S-BM, 100S-OP, and 108CHA data sets, with a root-mean-squared error (RMSE) between the observed and equicorrelation matrices of approximately 5%–10%, but less so for the four other data sets, with an RMSE of approximately 10%–20%. Thus, we can assess the performance of our optimal portfolio size across different correlation structures. Second, the parameters $ \left({\kappa}_{M,1},{\kappa}_{M,2},{\kappa}_{M,3}\right) $ defined in (10)–(12), which control the impact of the elliptical fat tails of returns on the EU, substantially depart from 1 (i.e., from normality). Thus, it is crucial to account for fat tails when implementing combination rules given their impact on combination coefficients.

B. Portfolio Strategies

We evaluate the performance of the 11 portfolio strategies listed in Table 2. The first three are the optimal combination rules in Section III.B and Appendix A.I: the 2F, 3FGMV, and 3FEW. We apply these strategies to all $ M $ assets or a subset of $ N $ assets, with $ N $ chosen optimally as $ {\hat{N}}_{2f}^{\star } $ in (18) for 2F, $ {\hat{N}}_{3f,g}^{\star } $ in (A.10) for 3FGMV, and $ {\hat{N}}_{3f, ew}^{\star } $ in (A.19) for 3FEW, estimated following Appendix A.II.

TABLE 2 List of Portfolio Strategies Considered in the Empirical Analysis

The next two portfolio strategies are individual portfolios: the EW portfolio and the SGMV portfolio in (A.1). We then consider two portfolio strategies that combine EW and SGMV with the risk-free asset to maximize the EU, EWRF and GMVRF, which might be preferred to 2F, 3FGMV, and 3FEW because they disregard the SMV portfolio that faces high estimation risk. We explain how we estimate the EWRF portfolio in Section OA.5 of the Supplementary Material. Regarding the GMVRF portfolio, we follow Kan and Lassance ((Reference Kan and Lassance2025), Section VI.C.2) and estimate it as

(24) $$ {\hat{\boldsymbol{w}}}_{gmvrf}=\frac{\kappa_{M,1}}{\kappa_{M,2}}\times \frac{T-M-2}{Tc_M}\times \frac{{\hat{\mu}}_{g,M}{\hat{\boldsymbol{\Sigma}}}^{-1}\mathbf{1}}{\gamma }, $$

where $ {\hat{\mu}}_{g,M} $ is defined in (A.27) and the estimation of $ {\kappa}_{M,1} $ and $ {\kappa}_{M,2} $ is described in Appendix A.II.A.

The next portfolio strategy we consider is the SMV portfolio, which we implement in three different ways. In the first case, we compute the SMV portfolio on all $ M $ assets using equation (7). In the second and third cases, we reduce the portfolio size with soft- and hard-thresholding using the SMV-ST and SMV-HT portfolios introduced in Section IV.C.Footnote ¹⁹

The last portfolio strategy we implement is the F+A strategy introduced in Section IV.C. We refer to Section OA.4 of the Supplementary Material for more details.

Finally, we consider three different estimates of the covariance matrix. First, the sample estimator in (5) for consistency with our theory. Second, the linear shrinkage estimator of Ledoit and Wolf (Reference Ledoit and Wolf2004). Third, the nonlinear shrinkage estimator of Ledoit and Wolf (Reference Ledoit and Wolf2020), for which we report the results in Section OA.7.2 of the Supplementary Material for conciseness.Footnote ²⁰

Note that whereas we implement the EWRF, SGMV, GMVRF, SMV, SMV-ST, SMV-HT, and F+A benchmark portfolios on all $ M $ assets, we study the impact of the portfolio size $ N $ on their empirical performance in Section OA.7.8 of the Supplementary Material.

C. Asset Selection Rules

Given a portfolio size $ N\le M $ , we consider several selection rules to decide which $ N $ assets to select. Our purpose is not to find an optimal asset selection rule. Instead, our main objective is to illustrate the added value of optimally reducing the portfolio size, and that even simple but sensible selection rules can deliver substantial gains over choosing all $ M $ assets.

For comparison purposes, we apply the proposed selection rules not just to 2F, 3FGMV, and 3FEW but also to the EW portfolio to evaluate their intrinsic value without being affected by estimation errors in optimized portfolio weights. In that case, we take $ N=T/2 $ , where $ T=120 $ months as explained in Section V.D, because the optimal $ N $ is close to $ T/2 $ for typical levels of equity correlations as explained in our theory and simulations.

We propose 11 asset selection rules, listed in Table 3, satisfying two criteria: They are simple to understand and implement, and apart from the random rule, they are sensible (i.e., they have an economic intuition). The first selection rule consists in choosing all assets and setting $ N=M $ . The 10 remaining selection rules select a subset of $ N\le M $ assets. Among those, the first one is a random selection, the next six are based solely on marginal information about the assets, and the last three rules also consider information about their dependence. Moreover, the first nine rules select the assets before computing the portfolio weights, whereas the last rule selects the subset of assets after optimizing the weights on all assets. We now describe these 10 asset selection rules.Footnote ²¹

TABLE 3 List of Asset Selection Rules Considered in the Empirical Analysis

Random selection (Rand). We randomly select the $ N $ assets among all $ M $ assets and report the performance across 100 repetitions of the out-of-sample analysis after merging all portfolio returns. This random rule serves as a benchmark to which other more sensible selection rules can be compared, and it is consistent with the simulation analysis of Section IV.

Maximum Sharpe ratio (MaxSR). We select the $ N $ assets with the maximum in-sample Sharpe ratios in the estimation window, because the EU of the SMV portfolio in (14) and the 2F in (16) both depend on the assets’ Sharpe ratios. This selection rule is expected to work particularly well when there is momentum in the assets.Footnote ²² We can anticipate several issues with this rule. First, it ranks the assets according to average returns that are notoriously noisy (Merton (Reference Merton1980)). Second, the selected assets have the best Sharpe ratios, and thus we might short good assets. Third, we do not include assets with the worst Sharpe ratios, although shorting them might deliver gains.

Minimum Sharpe ratio (MinSR). We select the $ N $ assets with the minimum in-sample Sharpe ratios. This rule will work well under mean reversion in the assets, or from using short positions. We expect MinSR to face the same issues as MaxSR, that is, not having access to the assets with the best Sharpe ratios to form long positions, and ending up with long positions on bad assets.

Best–worst Sharpe ratio (BWSR). We blend the MaxSR and MinSR rules by selecting the $ \left\lceil N/2\right\rceil $ assets with the best in-sample Sharpe ratios and the $ \left\lfloor N/2\right\rfloor $ assets with the worst in-sample Sharpe ratios, which can be combined with long and short positions. This is in line with the standard way of constructing anomaly portfolios as long-short portfolios of stocks in the top and bottom quantiles of a given firm characteristic. However, BWSR may lead to more extreme portfolio weights that do not generalize well out of sample.

The MaxSR, MinSR, and BWSR selection rules rank assets based on their in-sample Sharpe ratios.Footnote ²³ Given that average returns are notoriously noisy, these rankings are bound to be unstable.Footnote ²⁴ As a result, the set of selected assets is likely to change substantially from one period to another, increasing portfolio turnover and transaction costs. To address this point, the next three asset selection rules in Table 3 are the following:

Maximum, minimum, and best–worst variance (MaxVar, MinVar, BWVar). These selection rules rank assets based on the in-sample variance instead of Sharpe ratio. The resulting rankings of assets are expected to be more stable because the variance is quite persistent over time. The MaxVar rule selects assets with the maximum variances, which are bound to also have larger expected returns if a positive risk–return trade-off stands. However, the low-risk anomaly implies that assets and portfolios with lower variances often have larger expected returns (Ang, Hodrick, Xing, and Zhang (Reference Ang, Hodrick, Xing and Zhang2006), Moreira and Muir (Reference Moreira and Muir2017)), which the MinVar rule can exploit.

Minimum correlation with the first principal component (MinPC). We select the assets whose returns have minimum correlations with the first principal component (PC), which are least explained by the remaining assets and, thus, offer higher diversification potential.

Best portfolio Sharpe ratio (Best $ {\theta}_N^2 $ ). This selection rule follows Ao et al. (Reference Ao, Li and Zheng2019) and selects the assets that deliver a high maximum squared Sharpe ratio, $ {\theta}_N^2 $ in (2). It works in 3 steps. First, we form 1,000 random selections of $ N $ assets. Second, for each selection, we estimate $ {\theta}_N^2 $ , as detailed in Appendix A.II. Third, we pick the selection that corresponds to the 95% quantile of all estimated values of $ {\theta}_N^2 $ to avoid outliers.

Maximum absolute weights (MaxW). The above selection rules are independent of the portfolio strategy. However, the in-sample portfolio weights on all $ M $ assets are a valuable information. Therefore, MaxW selects the assets based on the portfolio strategy that will be applied to the selected assets. It works in 3 steps. First, compute $ \hat{\boldsymbol{w}}\in {\mathrm{\mathbb{R}}}^M $ , the portfolio weights on all $ M $ assets for a given strategy (2F, 3FGMV, or 3FEW). Second, select the $ N $ assets with the maximum absolute weights $ \mid {\hat{w}}_i\mid $ . Third, recompute the strategy on the $ N $ assets to obtain $ \hat{\boldsymbol{w}}\in {\mathrm{\mathbb{R}}}^N $ .Footnote ²⁵ MaxW selects the assets whose weights are large and, thus, identified as relevant. Note that MaxW does not apply to the EW portfolio whose weights are equal. One issue we anticipate is that MaxW inherits large estimation errors as it is based on portfolio weights optimized under a large $ M/T $ .

In Section OA.7.7 of the Supplementary Material, we introduce a measure that determines how much more or less overlap there is between two asset selection rules relative to a pair of independent random selection rules. For essentially all pairs of selection rules we consider, we find that there is either around the same or less overlap than under a random selection. That is, our selection rules select dissimilar assets, which allows us to test our portfolio strategies on different investment universes.

D. Out-of-Sample Methodology

We evaluate the out-of-sample portfolio performance with a standard monthly rebalancing. Specifically, at the end of month $ t $ , we estimate portfolio $ k $ over the $ T $ previous months, and we compute its out-of-sample return in month $ t+1 $ . We consider a fixed sample size of $ T=120 $ months.Footnote ²⁶ We repeat this process iteratively, resulting in $ {T}_{tot}-T $ out-of-sample gross returns $ {r}_{gross,k,t} $ , where $ {T}_{tot} $ is the total number of months in the data set. We then compute the net out-of-sample returns, $ {r}_{net,k,t}={r}_{gross,k,t} $ if $ t=T+1 $ and

(25) $$ {r}_{net,k,t}=\left(1+{r}_{gross,k,t}\right)\left(1-\kappa \times {\mathrm{turnover}}_{k,t-1}\right)-1\hskip1em \mathrm{if}\hskip1em t=T+2,\dots, {T}_{tot}, $$

where $ \kappa $ is the proportional transaction cost parameter and

(26) $$ {\mathrm{turnover}}_{k,t}=\sum \limits_{i=1}^N\mid {w}_{i,k,t}-{w}_{i,k,{\left(t-1\right)}^{+}}\mid, \hskip1em t=T+1,\dots, {T}_{tot}, $$

with $ {w}_{i,k,t} $ the weight of asset $ i $ in month $ t $ and $ {w}_{i,k,{\left(t-1\right)}^{+}} $ the prior-month weight before rebalancing in month $ t $ . We set $ \kappa =10 $ basis points in line with the average bid–ask spreads reported by Engle, Ferstenberg, and Russell (Reference Engle, Ferstenberg and Russell2012) and Frazzini, Israel, and Moskowitz (Reference Frazzini, Israel and Moskowitz2018). Finally, we compare the portfolio strategies in terms of annualized out-of-sample utility net of transaction costs,

(27) $$ {U}_k=12\times \left({\hat{\mu}}_k-\frac{\gamma }{2}{\hat{\sigma}}_k^2\right), $$

where $ {\hat{\mu}}_k $ and $ {\hat{\sigma}}_k^2 $ are the sample mean and variance of $ {r}_{net,k,t} $ . We set a risk-aversion coefficient of $ \gamma =1 $ .Footnote ²⁷ As explained in Section V.C, we repeat this out-of-sample analysis 100 times under the random asset selection rule and compute $ {U}_k $ after merging all portfolio returns.Footnote ²⁸ In Section OA.7.4 of the Supplementary Material, we also report the out-of-sample Sharpe ratio.

For 2F, 3FGMV, and 3FEW, the assets change as we re-estimate the optimal $ N $ and apply an asset selection rule. Because our methodology disentangles the determination of the optimal $ N $ from the choice of the $ N $ assets, we can also disentangle the rebalancing of portfolio weights from the asset selection. Therefore, to make our method less costly and more practical, we rebalance the portfolio weights every month but change the optimal $ N $ and selected assets every year.Footnote ²⁹

Finally, we compute two-sided $ p $ -values for the statistical test of the difference between the net utility of the 2F, 3FGMV, and 3FEW portfolios computed on all assets versus under each of the 10 other selection rules in Table 3. For SMV-HT and SMV-ST, we report $ p $ -values relative to SMV. To compute the $ p $ -values, we generate $ \mathrm{10,000} $ bootstrap samples using the stationary block bootstrap approach of Politis and Romano (Reference Politis and Romano1994) with an average block size of 5, and then use the methodology of Ledoit and Wolf ((Reference Ledoit and Wolf2008), Remark 3.2) to produce the $ p $ -values. We use the symbols ◯, ◑, and ● to indicate that the $ p $ -value is less than 10%, 5%, and 1%, respectively.

E. Discussion of Results

In Table 4, we report the annualized net out-of-sample utility, in percentage points, of the 11 portfolio strategies listed in Table 2. For the 2F, 3FGMV, 3FEW, and EW portfolio strategies, we report the performance across the 11 asset selection rules in Table 3.

TABLE 4 Annualized Net Out-of-Sample Utility in Empirical Data

The results reported in Table 4 show that portfolio strategies estimated with the shrinkage covariance matrix generally outperform those estimated with the sample one. Therefore, albeit our theory considers the sample covariance matrix for tractability, it is beneficial to shrink it. The ranking of portfolio strategies and asset selection rules is overall similar in both cases, and thus, unless stated otherwise, the discussion that follows essentially applies to either estimator.

First, we address our main objective, which is to assess the value of optimally reducing the portfolio size. For either of 2F, 3FGMV, and 3FEW, reducing the portfolio size delivers substantial performance gains. Specifically, the asset selection rule “All,” which sets $ N=M $ , is almost always outperformed by the remaining selection rules.Footnote ³⁰ The differences are often statistically significant, particularly under the shrinkage covariance matrix. Moreover, for the six data sets of characteristic and industry portfolios, the differences are systematically positive for the Rand, MinSR, BWSR, MaxVar, BWVar, and MinPC asset selection rules and are often substantial. For instance, under the shrinkage covariance matrix, on average across these six data sets, the 2F strategy on all $ M $ assets delivers an annualized net utility of 11.30 percentage points, versus 58.17 with the BWSR selection rule. Turning to the 100STO data set, asset selection rules based on average returns (MaxSR, MinSR, BWSR) do not work as well because there is less predictability in average returns among individual stocks. However, the MinVar and BWVar rules based on the variance, as well as the Rand rule, consistently outperform selecting all $ M $ assets. We depict these findings in Figure 8, which shows, for the shrinkage covariance matrix, the difference between the annualized net out-of-sample utility of 2F, 3FGMV, and 3FEW implemented on a subset of $ {\hat{N}}^{\star } $ assets versus all $ M $ assets. Having addressed our main objective, we now discuss other findings from Table 4.

FIGURE 8 Difference in Net Out-of-Sample Utility Relative to Investing in All Assets

Figure 8 depicts, for three different portfolio strategies, the difference between the annualized net out-of-sample utility in percentage points obtained when implementing the portfolios on a subset of $ N $ assets, where the optimal $ N $ is estimated using our theory, using 10 asset selection rules relative to the case where the portfolios are implemented on all $ M $ assets. The three portfolio strategies are the two-fund rule (blue), the GMV-three-fund rule (green), and the EW-three-fund rule (red). Each graph considers one of the 10 asset selection rules described in Table 3. This figure is constructed following the methodology described in Section V.D. We consider seven data sets described in Table 1. We estimate the portfolios with the linear shrinkage covariance matrix of Ledoit and Wolf (Reference Ledoit and Wolf2004). The net out-of-sample utility is computed using rolling windows, a sample size $ T=120 $ months, and proportional transaction costs of 10 basis points. The risk-aversion coefficient is $ \gamma =1 $ .

Second, we turn to the EW portfolio. Under the random asset selection rule, reducing $ N $ from $ M $ to $ T/2 $ is detrimental. This is because the EW portfolio faces no estimation risk and, thus, benefits from the diversification gains coming from a larger $ N $ . However, when the selection of the $ N $ assets is done using a sensible rule, such as MaxSR or MinVar, reducing $ N $ often improves the EU of the EW portfolio by removing undesirable assets. We also observe that when $ N=M $ , it is difficult for the 2F, 3FGMV, and 3FEW portfolios to outperform the EW portfolio. In particular, EW outperforms in five out of seven data sets. However, when reducing the portfolio size to $ N={\hat{N}}^{\star } $ with our theory, there are a number of asset selection rules for which 2F, 3FGMV, and 3FEW systematically outperform EW, particularly when we shrink the covariance matrix, no matter if EW is implemented on all $ M $ assets or on $ T/2 $ assets with the same asset selection rule. This key observation highlights the practical importance of our theoretically guided optimal portfolio size.

Third, for the six data sets of characteristic and industry portfolios, applying the MinSR and MaxVar asset selection rules to the 2F, 3FGMV, and 3FEW strategies delivers consistently positive utilities almost systematically larger than those of the seven other benchmarks. Focusing on the shrinkage covariance matrix, the BWSR, BWVar, MinPC, and Best $ {\theta}_N^2 $ selection rules also consistently deliver positive utilities generally above the benchmarks. Turning to the 100STO data set, under the shrinkage covariance matrix, the MaxSR, MinVar, and BWVar selection rules yield consistently positive net utilities generally above the benchmarks. This is particularly true when applying these selection rules to 3FGMV and 3FEW because, for individual stocks, adding an exposure to SGMV or EW is valuable given the lower predictability in expected returns. These results are remarkable because the EW portfolio, in particular, is difficult to outperform with MV portfolios when $ M/T $ is large, which is the case in our data sets where $ M/T $ ranges from $ 0.78 $ to $ 0.9 $ . These results are due to our optimal portfolio size because, under the “All” selection rule, the 2F, 3FGMV, and 3FEW portfolios underperform the EW portfolio in five out of seven data sets.

Fourth, we turn to the Sharpe ratio selection rules (MaxSR, MinSR, BWSR). Consider first the six data sets of characteristic and industry portfolios. For the EW portfolio, MaxSR consistently outperforms selecting all $ M $ assets. In contrast, MinSR and BWSR underperform. Specifically, on average across these six data sets, the EW portfolio of all assets delivers an annualized net out-of-sample utility of 5.94 percentage points, versus 6.93, 5.08, and 5.60 under MaxSR, MinSR, and BWSR, respectively. These results indicate momentum in characteristic and industry portfolios as an EW portfolio of assets with maximum in-sample Sharpe ratios outperforms the full EW portfolio. In contrast, for the 100STO data set, it is the MinSR and BWSR selection rules that deliver a better EW portfolio than that applied to all assets, indicating reversal in individual stocks.

Although MaxSR works well among characteristic and industry portfolios for constructing an EW portfolio, we find the opposite for the 2F, 3FGMV, and 3FEW portfolios. Specifically, MinSR consistently outperforms selecting all $ M $ assets and MaxSR, the latter being one of the worst selection rules. These results mean that when we use optimized weights, the performance gains do not originate from selecting assets with the best marginal performance, but assets that can be best exploited by the portfolio (e.g., with short positions for the MinSR rule). Finally, BWSR, which capitalizes on long and short positions by selecting both good and bad assets, achieves its objective as it consistently outperforms MaxSR and MinSR under the shrinkage covariance matrix.Footnote ³¹

Fifth, we turn to the variance asset selection rules (MaxVar, MinVar, BWVar). Consider first the six data sets of characteristic and industry portfolios. For the EW portfolio, MinVar consistently outperforms selecting all $ M $ assets. Specifically, on average across the six data sets, the EW portfolio of all assets delivers an annualized net out-of-sample utility of 5.94 percentage points, versus 5.47, 6.44, and 5.65 under MaxVar, MinVar, and BWVar, respectively. This finding indicates a low-risk anomaly in these six data sets. However, as noted previously, this does not necessarily mean that MinVar is also best for the 2F, 3FGMV, and 3FEW strategies. Indeed, Figure 8 shows that for these portfolios, MinVar delivers a worse performance overall than MaxVar and BWVar. This can be explained because assets with maximum variances selected by MaxVar and BWVar often have low average returns as the low-risk anomaly predicts, which the 2F, 3FGMV, and 3FEW portfolios can leverage using small or short positions.Footnote ³² The insights drawn from the 100STO data set are similar, with the asset selection rules favored when considering the EW portfolio (i.e., MaxVar and BWVar), underperforming the MinVar rule when turning to 2F, 3FGMV, and 3FEW.

Sixth, the MinPC asset selection rule, which accounts for asset dependence by selecting those least correlated with the first PC, performs consistently well. In particular, under the shrinkage covariance matrix, the resulting 2F, 3FGMV, and 3FEW portfolios have positive net out-of-sample utilities larger than those under the “All” selection rule in all cases but one.

Seventh, the Best $ {\theta}_N^2 $ selection rule, which maximizes the portfolio Sharpe ratio, outperforms investing in all assets for the 2F, 3FGMV, and 3FEW portfolios under the shrinkage covariance matrix, in all cases but one.

Eighth, the MaxW selection rule outperforms on average all other rules under the shrinkage covariance matrix. For instance, considering the 3FGMV portfolio, MaxW delivers an annualized net utility of 72.15 percentage points on average across data sets, versus 55.09 for the second-best selection rule, BWSR. However, MaxW underperforms all other selection rules under the sample covariance matrix. This can be explained because, in that case, 2F, 3FGMV, and 3FEW are more subject to estimation risk, and thus selecting assets with large absolute weights is less desirable.

Finally, we consider the SMV, SMV-ST, SMV-HT, and F+A benchmarks. First, as expected, SMV is by far the worst strategy. Second, F+A improves upon SMV but is the second-worst strategy. This suggests that although the Da et al. (Reference Da, Nagel and Xiu2024) method is optimal under specific assumptions as $ N $ and $ T $ get large, it does not work as well as competitors in a finite-sample setting, in line with Figure 7. Third, SMV-ST, which maximizes the MV utility subject to an $ {L}_1 $ -norm constraint, greatly improves upon SMV and SMV-HT. However, in most cases, it underperforms the size-optimized 2F, 3FGMV, and 3FEW portfolios. Fourth, SMV-HT, which keeps only the assets whose SMV portfolio weights are above a given threshold, outperforms the plain SMV but delivers negative net utilities. In Section OA.7.3 of the Supplementary Material, we compare the portfolio size obtained with SMV-ST and SMV-HT to that obtained with our theory, $ {\hat{N}}_{smv}^{\star } $ . We find that the three methods set a small $ N $ on average and that our $ {\hat{N}}_{smv}^{\star } $ has much less variability across windows.