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On the optimal combination of naive and mean-variance portfolio strategies. (2022). Vrins, Frederic ; Vanderveken, Rodolphe ; Lassance, Nathan.
In: LIDAM Discussion Papers LFIN.
RePEc:ajf:louvlf:2022006.

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  1. . (IA114) The EU of the optimal portfolio combination of Kan and Zhou (2007) in (IA21) is then obtained by plugging κkz 1 and κkz 2 given by (IA20) in the EU formula (IA114).
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  2. (IA28) We evaluate the empirical performance of this mixed strategy in Section IA.3. IA.2.4 The optimal four-fund combination rule Kan and Zhou (2007) combine the SMV portfolio with the SGMV portfolio, while as in Tu and Zhou (2011) we combine the SMV portfolio with the EW portfolio. As discussed in IA9 Section 5.3, we find empirically in Tables 3 and 4 that it is generally preferable to combine with the EW portfolio when the sample size T = 120, and with the SGMV portfolio when the sample size T = 240. In this section, we derive the optimal four-fund portfolio that combines the SMV portfolio with both the SGMV and EW portfolios. The resulting four-fund portfolio combination is ŵ(κ) = κ1 γ b Σ−1 ̂ + κ2wew + κ3 γ b
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  3. 2. The fully invested strategy of Kan, Wang, and Zhou (2021) delivers a larger expected out-of-sample utility than the estimated optimal three-fund strategy of Kan and Zhou (2007), EU(ŵ(κkwz )) ≥ EU ŵ(κkz 1 , κ̂kz 2 ) , if and only if b ≥ 0 and γ belongs to the following interval: γ ∈ " γtan c1 √ b c1σg # , (IA25) where b = θ2 g + c1 − θ2 + ψ2 g T − N − 2 T − N − 1 κkwz + dκkz 1 + (c − 1)γtanκkz 2 + c T (1/c2 − (κkz 1 )2 ) ! . (IA26) For the 25SBTM dataset and T = 120, we find that the interval is γ ∈ [3.18 1.73]. Those investors are better off not investing in the risk-free asset besides the fully invested SMV and SGMV portfolios.
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  16. DeMiguel, V., A. Martín-Utrera, and F. Nogales. 2013. Size matters: Optimal calibration of shrinkage estimators for portfolio selection. Journal of Banking and Finance 37:3018–34.

  17. DeMiguel, V., L. Garlappi, and R. Uppal. 2009. Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? The Review of Financial Studies 22:1915–53.

  18. DeMiguel, V., L. Garlappi, F. Nogales, and R. Uppal. 2009. A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Management Science 55:798–812.

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  23. Σ−1 (̂ − ̂g1). (IA16) is a zero-cost portfolio (1′ ŵz = 0). To ensure that ŵ(κ) is fully invested in risky assets, the authors impose the convexity constraint κ1 + κ2 = 1, so that the combination depends on a single combination coefficient κ: ŵ(κ) = ŵg + κŵz. (IA17) IA6 Kan, Wang, and Zhou (2021) show that the optimal combination coefficient κ is κkwz = (T − N)(T − N − 3) (T − 2)(T − N − 2) ψ2 g
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  24. ψ2 g + (N − 1)/T . (IA18) Relaxing the convexity constraint would amount to combine the SMV and SGMV portfolios with the risk-free asset, which is what the three-fund rule of Kan and Zhou (2007) is designed to do. Their optimal portfolio combination is ŵ(κkz ) = κkz 1 γ b Σ−1 ̂ + κkz 2 γ b Σ−1 1 (IA19) with κkz 1 = 1 c ψ2 g
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  25. ψ2 g + N/T and κkz 2 = g(1/c − κkz 1 ). (IA20) In the next proposition, we derive the EU of the optimal portfolio rules of Kan and Zhou (2007) and Kan, Wang, and Zhou (2021).
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  26. IA.2.2 Investing in the risk-free asset can hurt performance We observe in our empirical results in Tables 3 and 4 that the optimal combination of the SMV and SGMV portfolios without a risk-free asset in Kan, Wang, and Zhou (2021) often outperforms the combination with a risk-free asset in Kan and Zhou (2007) when the riskaversion coefficient γ is rather small (γ = 3 and 5). In this section, we show this can be explained due to the convexity constraint, similar to the results in Sections 4 and IA.2.1. Kan, Wang, and Zhou (2021) combine the fully invested SMV and SGMV portfolios, which corresponds to the portfolio combination ŵ(κ) = κ1ŵg + κ2(ŵg + ŵz), (IA15) where ŵg = b Σ−1 1/(1′ b Σ−1 1) is the SGMV portfolio and ŵz = 1 γ b
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  27. IA22 IA.4 Proofs of all results We make extensive use of three properties throughout the proofs. First, because returns are assumed iid normal, the sample mean is distributed as ̂ ∼ N(, Σ/T), the sample covariance matrix is distributed as (T −N −2) b Σ ∼ WN (T −1, Σ), and they are independent of each other. Second, the inverse sample covariance matrix is unbiased because of the 1/(T − N − 2) coefficient in (6), E[ b Σ−1 ] = Σ−1 . Third, the expectation of a quadratic form in the random variable x is E[x′ Ax] = E[x]′ AE[x] + Trace(AV[x]), (IA47) where A is a constant matrix (Rencher and Schaalje, 2008).
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  28. IA43 References in Internet Appendix Ao, M., Y. Li, and X. Zheng. 2019. Approaching mean-variance efficiency for large portfolios.
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  32. Just like the maximum utility in (3) is unattainable in the presence of parameter uncertainty, the maximum Sharpe ratio θ = √ ′Σ−1 is also unattainable. To quantify the impact of parameter uncertainty on the out-of-sample Sharpe ratio of an estimated portfolio ŵ, we define the expected out-of-sample Sharpe ratio (ESR) as in DeMiguel, Martín-Utrera, and Nogales (2013): ESR(ŵ) = E[ŵ′ ] q E[ŵ′Σŵ] . (IA39) From the proof of Proposition 1, we have that the ESR of the combination between the SMV and EW portfolios in (9) is ESR(ŵ(κ)) = κ1 γ θ2 + κ2ew r κ2 1
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  48. Notes. This figure depicts the difference between the expected out-of-sample utility of: (i) the optimal threefund and two-fund rules (solid red), and (ii) the optimal four-fund and three-fund rules (dotted blue). The two-fund rule is that in Kan and Zhou (2007) that combines the sample tangent portfolio with the riskfree asset. The three-fund rule adds the equally weighted portfolio, and the four-fund rule adds the sample global-minimum-variance portfolio. The figure is constructed by calibrating the population vector of means and covariance matrix of stock excess returns from monthly returns on the 25 portfolios of stocks sorted on size and book-to-market spanning July 1926 to December 2021, and using a sample size T = 120. The differences are depicted as a function of the risk-aversion coefficient γ between 0.5 and 20. 3. The expected out-of-sample utility of the optimal four-fund portfolio is EU(ŵ(κopt )) = U⋆ − d
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  49. Okhrin, Y., and W. Schmid. 2006. Distributional properties of portfolio weights. Journal of Econometrics 134:235–56.

  50. Pflug, G., A. Pichler, and D. Wozabal. 2012. The 1/N investment strategy is optimal under high model ambiguity. Journal of Banking and Finance 36:410–7.

  51. Proof of Proposition IA.1 Part 1. The SMV portfolio ŵ⋆ is unbiased, and the SGMV portfolio ŵg too (Okhrin and Schmid, 2006). Therefore, the expected out-of-sample mean return of the portfolio combination (IA1) is E[ŵ(κ)] = κ1 γ θ2 + κ2g. (IA79) The expected out-of-sample variance decomposes as E[ŵ(κ)′ Σŵ(κ)] = κ2 1 γ2 (θ2 + d) + κ2 2E h ŵ′ gΣŵg i + 2κ1κ2 γ E h ̂′ b Σ−1 Σŵg i
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  52. Proposition IA.4. Let the combination coefficient κkz 2 be estimated by κ̂kz 2 in (IA23). Then, 1. The expected out-of-sample utility of the estimated optimal three-fund strategy of Kan and Zhou (2007) is EU(ŵ(κkz 1 , κ̂kz 2 )) = EU(ŵ(κkz )) − c 2γT (1/c2 − (κkz 1 )2 ), (IA24) where EU(ŵ(κkz )) in (IA21) is the utility when κkz 2 is known.
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  53. Rencher, A., and G. B. Schaalje. 2008. Linear Models in Statistics (2nd ed.). New Jersey: John Wiley & Sons.
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  54. Simaan, M., and Y. Simaan. 2019. Rational explanation for rule-of-thumb practices in asset allocation. Quantitative Finance 19:2095–109.

  55. Table IA.1 reports the annualized net out-of-sample Sharpe Ratio for all datasets listed in Table 1 and portfolio strategies listed in Table 2. We only report the results when using the sample covariance matrix for conciseness; the conclusions are consistent when using the shrinkage covariance matrix of Ledoit and Wolf (2004). In line with the theoretical predictions in Section IA.2.6, we observe that the constrained strategy of Tu and Zhou (2011) (TZ3F) outperforms the optimal strategy (OPT3F) in most cases, although the difference is small.
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  56. The mixed strategy (MIX3F) approaches the Sharpe ratio of the constrained strategy, but not quite due to estimation errors in the interval (30). Just like for the EU in Table 3, the proposed OPT3F strategy consistently outperforms the optimal two-fund rule of Kan and Zhou (2007) (KZ2F). The optimal combination of the SMV and SGMV portfolios in Kan and Zhou (2007) (KZ3F) performs similarly to OPT3F when the sample size T = 120, but consistently outperforms for T = 240. This means that, in terms of Sharpe ratio, combining SMV with SGMV is preferable to combining SMV with EW. Finally, the different combination strategies largely outperform the two naive benchmarks, EWRF and GMVRF, except for the 30IND dataset where EWRF achieves the best performance for T = 120 and the second-best performance for T = 240.
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  57. The Review of Financial Studies 32:2890–919.
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  58. Tu, J., and G. Zhou. 2011. Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics 99:204–15.

  59. Tu, J., and G. Zhou. 2011. Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies. Journal of Financial Economics 99:204–15. IA44
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  60. Yan, C., and H. Zhang. 2017. Mean-variance versus naïve diversification: The role of mispricing. Journal of International Financial Markets, Institutions and Money 48:61–81.
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  61. Yen, Y.-M. 2016. Sparse weighted-norm minimum variance portfolios. Review of Finance 20:1259–87.

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  40. Optimal Portfolio Using Factor Graphical Lasso. (2020). Seregina, Ekaterina ; Lee, Tae-Hwy.
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  41. Optimizing tail risks using an importance sampling based extrapolation for heavy-tailed objectives. (2020). Murthy, Karthyek ; Deo, Anand.
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  42. Relative utility bounds for empirically optimal portfolios. (2020). Rokhlin, Dmitry B.
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  44. A Framework for Online Investment Algorithms. (2020). Gebbie, Tim ; van Zyl, Terence ; Paskaramoorthy, Andrew.
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  45. Computational approaches and data analytics in financial services: A literature review. (2019). Zopounidis, Constantin ; Pardalos, Panos M ; Doumpos, Michalis ; Andriosopoulos, Dimitris.
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  46. Near-Optimal Bayesian Ambiguity Sets for Distributionally Robust Optimization. (2019). Gupta, Vishal.
    In: Management Science.
    RePEc:inm:ormnsc:v:65:y:2019:i:9:p:4242-4260.

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