Regularized Maximum Diversification Investment Strategy †
Abstract
:1. Introduction
2. The Model
- (1)
- remains bounded when
- (2)
- for any ,
3. The Regularized Portfolio
3.1. Tikhonov Regularization (TH)
3.2. The Spectral Cut-Off (SC)
3.3. Landweber–Fridman Regularization (LF)
4. Asymptotic Properties of the Selected Portfolio
4.1. Efficiency of the Regularized Diversified Portfolio
- Assumption A:
- is a trace class operator.
4.2. Data-Driven Method for Selecting the Tuning Parameter
5. Simulations and Empirical Study
5.1. Data
5.2. Simulation
- The sample-based diversified portfolio (SbDP). This strategy is obtained using sample moments to estimate the unknown parameters in the maximum diversification portfolio.
- The most diversified portfolio (MDP) proposed by Choueifaty et al. (2013). This strategy is obtained by solving the optimization problem in Equation (2) under the following constraint,The closed form associated with this new optimization problem is given as follows,
- The global minimum variance portfolio (GMVP) obtained by minimizing the variance of the return on the optimal selected portfolio. By solving this optimization problem, the following closed form is obtained,This solution is then estimated by replacing the covariance matrix by the sample covariance matrix.
- The regularized strategies such as: the ridge regularized diversified portfolio (RdgDP), the spectral cut regularized diversified portfolio (SCDP), and the Landweber–Fridman regularized diversified portfolio (LFDP).
- The equal-weighted portfolio which is also called the naive portfolio (XoNP) which allocates a constant amount 1/N+1 in each asset.
- The target (or the maximum Sharpe ratio) portfolio (TgP). The closed form of the target portfolio isThis portfolio is also estimated using sample moments such as the sample mean and the sample covariance matrix to estimate the unknown parameters.
- The linear factor-based shrinkage estimators proposed by Ledoit and Wolf (2003) (LWP). It consists of replacing the sample covariance matrix in the selected portfolio by an optimally weighted average of two existing estimators: the sample covariance matrix and single-index covariance matrix. This method involves also a tuning parameter that is unknown and has been selected by the authors. The tuning parameter selection procedure proposed in Ledoit and Wolf (2003) is based on minimizing the distance between the population covariance matrix and the regularized one. This implies that the way they select the turning parameter is different from our data-driven method. Therefore, the LWP will be considered here as a very good benchmark (and it will be the only benchmark that we consider) to evaluate the ability of our data-driven method to deliver additional performance compare to other data-driven methods.
5.3. Empirical Study
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Proposition 1
Appendix B. Proof of Proposition 2
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1. | The objective is to reduce the effect of estimation error on the performance of selected maximum diversification portfolio. |
Parameters for Factors Loadings | Parameters for Factors Returns | ||||||
---|---|---|---|---|---|---|---|
μb | Σb | μf | Σf | ||||
1.0267 | 0.0422 | 0.0388 | 0.0115 | 0.0063 | 0.0020 | 0.0003 | −0.0004 |
0.0778 | 0.0388 | 0.0641 | 0.0162 | 0.0011 | 0.0003 | 0.0009 | −0.0003 |
0.2257 | 0.0115 | 0.0162 | 0.0862 | 0.0028 | −0.0004 | −0.0003 | 0.0009 |
Strategies | Number of Risky Assets | ||||||
---|---|---|---|---|---|---|---|
10 | 20 | 40 | 60 | 80 | 90 | 100 | |
SbDP | 0.1549 | 0.0906 | 0.0889 | 0.0779 | 0.0652 | 0.0719 | 0.0704 |
XoNP | 0.2604 | 0.2604 | 0.2415 | 0.2525 | 0.2406 | 0.2461 | 0.2467 |
GMVP | 0.2227 | 0.2338 | 0.2098 | 0.2298 | 0.1710 | 0.1640 | 0.1449 |
MDP | 0.2514 | 0.2545 | 0.2410 | 0.2544 | 0.1778 | 0.1821 | 0.1935 |
TgP | 0.2608 | 0.2818 | 0.2662 | 0.2687 | 0.2026 | 0.1925 | 0.1699 |
LWP | 0.2589 | 0.2702 | 0.2688 | 0.2704 | 0.2628 | 0.2521 | 0.2507 |
RdgDP | 0.2587 | 0.2785 | 0.2817 | 0.2907 | 0.2947 | 0.2830 | 0.2991 |
SCDP | 0.2592 | 0.2872 | 0.2993 | 0.2898 | 0.2746 | 0.2887 | 0.2853 |
LFDP | 0.2605 | 0.2765 | 0.2840 | 0.2870 | 0.2850 | 0.2912 | 0.2980 |
True SR | 0.2626 | 0.2922 | 0.3393 | 0.3379 | 0.3592 | 0.3477 | 0.3657 |
Strategies | Number of Risky Assets | |||||||
---|---|---|---|---|---|---|---|---|
150 | 250 | 400 | 550 | 700 | 850 | 950 | 999 | |
SbDP | 0.1230 | 0.1104 | 0.103 | 0.0998 | 0.060 | 0.03 | 0.012 | 0.008 |
XoNP | 0.2630 | 0.2640 | 2507 | 0.240 | 0.238 | 0.2207 | 0.2180 | 0.220 |
GMVP | 0.3080 | 02908 | 0.2890 | 0.2780 | 0.250 | 0.1980 | 0.1017 | 0.095 |
MDP | 0.3280 | 0.3305 | 0.3198 | 0.309 | 0.2679 | 0.2892 | 0.1985 | 0.120 |
TgP | 0.3290 | 0.3105 | 0.307 | 0.3100 | 0.2608 | 0.210 | 0.180 | 0.098 |
LWP | 0.3302 | 0.3408 | 0.3318 | 0.3070 | 0.415 | 0.4504 | 0.4601 | 0.4807 |
RdgDP | 0.3702 | 0.3850 | 0.3980 | 0.458 | 0.524 | 0.540 | 0.558 | 0.601 |
SCDP | 0.3689 | 0.3860 | 0.3980 | 0.460 | 0.5230 | 0.535 | 0.590 | 0.608 |
LFDP | 0.3704 | 0.3840 | 0.3984 | 0.4560 | 0.5250 | 0.538 | 0.585 | 0.595 |
True SR | 0.3758 | 0.3904 | 0.407 | 0.489 | 0.5480 | 0.588 | 0.608 | 0.618 |
Strategies | Number of Risky Assets | |||||||
---|---|---|---|---|---|---|---|---|
150 | 250 | 400 | 550 | 700 | 850 | 950 | 999 | |
SbDP | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
XoNP | 0.004 | 0.002 | 0.007 | 0.005 | 0.000 | 0.000 | 0.000 | 0.000 |
GMVP | 0.008 | 0.004 | 0.006 | 0.007 | 0.000 | 0.000 | 0.000 | 0.000 |
MDP | 0.003 | 0.001 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
TgP | 0.009 | 0.003 | 0.008 | 0.004 | 0.001 | 0.000 | 0.008 | 0.000 |
LWP | 0.089 | 0.013 | 0.001 | 0.012 | 0.035 | 0.003 | 0.043 | 0.008 |
Strategies | Number of Risky Assets | |||||||
---|---|---|---|---|---|---|---|---|
150 | 250 | 400 | 550 | 700 | 850 | 950 | 999 | |
SbDP | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
XoNP | 0.003 | 0.001 | 0.008 | 0.007 | 0.001 | 0.000 | 0.000 | 0.000 |
GMVP | 0.010 | 0.003 | 0.007 | 0.002 | 0.001 | 0.000 | 0.000 | 0.000 |
MDP | 0.005 | 0.001 | 0.004 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
TgP | 0.008 | 0.004 | 0.005 | 0.004 | 0.002 | 0.000 | 0.008 | 0.000 |
LWP | 0.090 | 0.014 | 0.003 | 0.009 | 0.040 | 0.007 | 0.001 | 0.007 |
Strategies | Number of Risky Assets | |||||||
---|---|---|---|---|---|---|---|---|
150 | 250 | 400 | 550 | 700 | 850 | 950 | 999 | |
SbDP | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
XoNP | 0.004 | 0.003 | 0.006 | 0.005 | 0.000 | 0.000 | 0.000 | 0.000 |
GMVP | 0.020 | 0.003 | 0.005 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 |
MDP | 0.003 | 0.002 | 0.003 | 0.002 | 0.001 | 0.000 | 0.000 | 0.000 |
TgP | 0.003 | 0.002 | 0.004 | 0.002 | 0.001 | 0.000 | 0.001 | 0.000 |
LWP | 0.104 | 0.043 | 0.002 | 0.008 | 0.032 | 0.004 | 0.002 | 0.006 |
Strategies | Number of Risky Assets | ||||||
---|---|---|---|---|---|---|---|
10 | 20 | 40 | 60 | 80 | 90 | 100 | |
SbDP | 2.315 | 2.307 | 2.304 | 2.08 | 1.308 | 1.128 | 1.098 |
XoNP | 3.103 | 3.140 | 3.180 | 3.184 | 3.325 | 3.288 | 3.154 |
GMVP | 3.242 | 3.241 | 3.150 | 3.185 | 3.147 | 3.155 | 3.093 |
MDP | 3.252 | 3.320 | 3.240 | 3.290 | 3.320 | 3.265 | 3.254 |
TgP | 3.240 | 3.170 | 3.105 | 3.050 | 3.132 | 3.149 | 3.080 |
LWP | 3.345 | 3.360 | 3.320 | 3.380 | 3.398 | 3.403 | 3.420 |
RdgDP | 3.325 | 3.428 | 3.480 | 3.590 | 3.598 | 3.602 | 3.640 |
SCDP | 3.347 | 3.435 | 3.446 | 3.570 | 3.589 | 3.615 | 3.625 |
LFDP | 3.289 | 3.405 | 3.470 | 3.548 | 3.604 | 3.509 | 3.638 |
True DR | 3.45 | 3.56 | 3.57 | 3.68 | 3.8 | 3.7 | 3.9 |
Strategies | XoNP | GMVP | MDP | TGP | RdgP | LFP | SCP | LWP | |
---|---|---|---|---|---|---|---|---|---|
FF30 | ER | 0.0110 | 0.01134 | 0.0121 | 0.017 | 0.0149 | 0.014 | 0.014 | 0.014 |
V | 0.0540 | 0.0630 | 0.058 | 0.076 | 0.063 | 0.057 | 0.061 | 0.067 | |
SR | 0.204 | 0.180 | 0.209 | 0.224 | 0.237 | 0.246 | 0.2295 | 0.209 | |
FF100 | ER | 0.0103 | 0.0127 | 0.015 | 0.0173 | 0.0200 | 0.0201 | 0.0203 | 0.019 |
V | 0.0485 | 0.075 | 0.088 | 0.091 | 0.0772 | 0.0770 | 0.078 | 0.082 | |
SR | 0.212 | 0.1693 | 0.1705 | 0.1901 | 0.2590 | 0.2610 | 0.2602 | 0.2317 |
P | EW | Strategies | |||||||
---|---|---|---|---|---|---|---|---|---|
SbDP | GMVP | MDP | TgP | LWP | RdgDP | SCDP | LFDP | ||
FF30 | 60 | 6.890 | 4.329 | 2.809 | 4.209 | 1.0328 | 0.9952 | 0.989 | 0.9872 |
120 | 5.605 | 3.901 | 2.087 | 3.290 | 0.9892 | 0.7140 | 0.7203 | 0.6450 | |
FF100 | 120 | 9.789 | 6.2390 | 5.978 | 6.309 | 1.7808 | 1.3267 | 1.3890 | 1.2078 |
240 | 7.089 | 4.297 | 3.879 | 4.2870 | 1.3065 | 1.0349 | 1.0398 | 1.096 |
Strategies | Market | HML | SMB | Intercept |
---|---|---|---|---|
Rdg-regularized Portfolio | 0.9168 (0.000) | 0.079 (0.531) | −0.139 (0.302) | 0.0075 (0.057) |
LF- regularized Portfolio | 0.823 (0.000) | 0.174 0.153) | −0.1651 (0.204) | 0.0125 (0.001) |
SC-regularized Portfolio | 1.02 (0.000) | −0.127 (0.177) | −0.133 (0.189) | 0.0077 (0.010) |
Most-Diversified Portfolio | 0.72 (0.000) | 0.13 (0.344) | 0.098 (0.506) | 0.007 (0.002) |
Equal-Weight-Portfolio | 1.002 (0.000) | 0.5104 (0.000) | 0.33 (0.000) | 0.0001 (0.815) |
Global-Minimum-Variance Portfolio | 0.416 (0.000) | −0.125 (0.319) | 0.155 (0.247) | 0.0094 (0.000) |
Target-Portfolio | 0.43 (0.000) | 0.144 (0.367) | 0,207 (0.226) | 0.010 (0.000) |
LW-Portfolio | 0.802 (0.000) | 0.074 (0.247) | 0.207 (0.226) | 0.0082 (0.067) |
Strategies | Market | HML | SMB | Intercept |
---|---|---|---|---|
Rdg-regularized Portfolio | 1.03 (0.000) | 0.24 (0.003) | 0.36 (0.000) | 0.0007 (0.767) |
LF- regularized Portfolio | 0.93 (0.000) | 0.22 (0.003) | 0.25 (0.001) | 0.0046 (0.042) |
SC-regularized Portfolio | 0.86 (0.000) | 0.27 (0.000) | 0.21 (0.031) | 0.0054 (0.053) |
Most-Diversified Portfolio | 0.46 (0.000) | −0.285 (0.000) | 0.070 (0.391) | 0.002 (0.001) |
Equal-Weight-Portfolio | 0.983 (0.000) | 0.061 (0.006) | 0.265 (0.000) | 0.0013 (0.050) |
Global-Minimum-Variance Portfolio | 0.46 (0.000) | −0.146 (0.008) | 0.077 (0.188) | 0.0021 (0.017) |
Target-Portfolio | 0.54 (0.000) | −0.44 (0.000) | −0.21 (0.019) | 0.013 (0.000) |
LW-Portfolio | 0.982 (0.000) | 0.272 (0.0098) | 0.4112 (0.0301) | 0.0006 (0.429) |
P | EW | Strategies | |||||||
---|---|---|---|---|---|---|---|---|---|
SbDP | GMVP | MDP | TgP | LWP | RdgDP | SCDP | LFDP | ||
100 A | 120 | 0.0850 | 0.1506 | 0.2458 | 0.1983 | 0.3702 | 0.4382 | 0.4380 | 0.4397 |
240 | 0.0982 | 0.1604 | 0.260 | 0.2028 | 0.3809 | 0.4565 | 0.4567 | 0.4578 | |
150 A | 180 | 0.0750 | 0.1204 | 0.309 | 0.1407 | 0.4108 | 0.5353 | 0.5320 | 0.5462 |
240 | 0.0895 | 0.1750 | 0.320 | 0.1890 | 0.4208 | 0.5603 | 0.5609 | 0.5579 |
Assets | EW | Strategies | |||||||
---|---|---|---|---|---|---|---|---|---|
SbDP | GMVP | MDP | TgP | LWP | RdgDP | SCDP | LFDP | ||
100 Assets | 120 | 9.450 | 6.786 | 4.675 | 6.679 | 3.348 | 2.1067 | 2.0801 | 2.0682 |
240 | 6.978 | 5.308 | 3.892 | 5.234 | 3.078 | 1.491 | 1.608 | 1.569 | |
150 Assets | 180 | 10.489 | 7.345 | 6.782 | 7.328 | 3.897 | 2.678 | 2.780 | 2.8960 |
240 | 8.0789 | 5.542 | 4.032 | 5.438 | 3.057 | 2.104 | 2.0978 | 2.0956 |
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Koné, N. Regularized Maximum Diversification Investment Strategy. Econometrics 2021, 9, 1. https://doi.org/10.3390/econometrics9010001
Koné N. Regularized Maximum Diversification Investment Strategy. Econometrics. 2021; 9(1):1. https://doi.org/10.3390/econometrics9010001
Chicago/Turabian StyleKoné, N’Golo. 2021. "Regularized Maximum Diversification Investment Strategy" Econometrics 9, no. 1: 1. https://doi.org/10.3390/econometrics9010001
APA StyleKoné, N. (2021). Regularized Maximum Diversification Investment Strategy. Econometrics, 9(1), 1. https://doi.org/10.3390/econometrics9010001