# Target Matrix Estimators in Risk-Based Portfolios

## Abstract

**:**

## 1. Introduction

## 2. Risk-Based Portfolios

## 3. Shrinkage Estimator

#### 3.1. Target Matrix Literature Review

#### 3.2. Estimators for the Target Matrix

#### 3.3. The Impact of Misspecification in the Target Matrix

## 4. Case Study—Monte Carlo Analysis

#### 4.1. Main Results

#### 4.1.1. Results on Portfolio Weights

#### 4.1.2. Sensitivity to Shrinkage Intensity

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | The majority of papers on risk-based portfolios are published in journal aimed at practitioners, as the Journal of Portfolio Management. |

2 | With this we refer to the population covariance matrix, which by definition is not observable and then unfeasible. Hence, $\mathrm{\Sigma}$ is estimated taking into account the observations stored in X: we will deeply treat this in the next section. |

3 | The sample covariance matrix is the Maximum Likelihood Estimator (MLE) under Normality, therefore it lets data speaks without imposing any structure. |

4 | In reality, we exclude the Scaled Identity of De Miguel et al. (2013) because of its great similarity with the Identity and Variance Identity implemented in our study. |

5 | Ardia et al. (2017) imposes Asset-1 and Asset-2 to have 10% annual volatility; Asset-3 to have 20% annual volatility; correlations between Asset-1/Asset-2 and Asset-1/Asset-3 are set as negative and correlation between corporate bonds and equities (Asset-2/Asset-3) is set as positive. However, to better resemble real data, specifically the S&P500, the US corporate index and the US Treasury Index total returns, we assume all three correlation parameters to be positive. |

6 | |

7 | Simulations were done in MATLAB setting the random seed generator at its default value, thus ensuring the full reproducibility of the analysis. Related code available at the GitHub page of the author: https://github.com/marconeffelli/Risk-Based-Portfolios. |

**Figure 1.**Frobenius norm between true and estimated weights; first row reports misspecification in volatility, while second row in correlation. The surfaces’ three dimensions are: the shrinkage intensity in y axis (from 0 to 1); the misspecification in the volatility (from 0 to 0.5) or in the correlation (from 0 to 1) in x axis and the Frobenius norm in z axis. Each column refers to a specific risk-based portfolio. From the left to the right: Minimum Variance (MV), Inverse Volatility (IV), Equal-Risk-Contribution (ERC), Maximum Diversification (MD), respectively.

**Figure 2.**The reciprocal 1-norm condition number (y-axis) as the $p/n$ ratio moves from $\frac{p}{60}$ to $\frac{p}{6000}$ (x-axis). Each column corresponds to a specific target matrix: from left to right, the Identity (Id): blue circle-shaped; the Variance Identity (VId): green square-shaped; the Single-Index (SI): red hexagram-shaped; the Common Covariance (CV): black star-shaped; the Constant Correlation (CC): cyan plus-shaped; and the Exponential Weighted Moving Average (EWMA): magenta diamond-shaped, respectively. Each row corresponds to a different $p$: in ascendant order from 10 (first row) to 100 (third row).

**Figure 3.**Surfaces representing the Frobenius norm (z-axis) between the true and the estimated target matrices, considering the shrinkage intensity (y-axis) and the $p/n$ ratio (x-axis). Each column corresponds to a specific target matrix: from left to right, the Identity (Id), the Variance Identity (VId), the Single-Index (SI), the Common Covariance (CV), the Constant Correlation (CC), and the EWMA, respectively. Each row corresponds to a different $p$: in ascendant order from $p=10$ (first row) to $p=100$ (third row).

**Figure 4.**Optimal shrinkage intensity parameters for which the Frobenius norm is minimized. Each column corresponds to a specific target matrix: from left to right, the Identity, the Variance Identity, the Single-Index, the Common Covariance, the Constant Correlation, and the EWMA, respectively. Each row corresponds to a different $n$: in ascendant order from $n=60$ (first row) to $p=6000$ (fifth row). In each subplot, the MV portfolio is blue circle-shaped; the IV is green-square shaped; the ERC is red-triangle shaped; and the MD is black-cross shaped.

**Figure 5.**Frobenius norm for portfolio weights with regard to the shrinkage intensity parameter, when $p=100$.

**Table 1.**Literature review of target matrices. “SCVm” = sample covariance matrix. “N.A.” = not available. “GMVP” = Global Minimum Variance Portfolio.

Reference | Matrix to Shrink | Target Matrix | Shrinkage Intensity | Portfolio Selection Rule | Research Question |
---|---|---|---|---|---|

(Ledoit and Wolf 2003) | SCVm | Market Model and Variance Identity | Risk-function minimisation | Classical Markowitz problem | Portfolio Performance comparison |

(Ledoit and Wolf 2004a) | SCVm | Identity | Risk-function minimisation | N.A. | Theoretical paper to gauge the shrinkage asymptotic properties |

(Ledoit and Wolf 2004b) | SCVm | Constant Correlation Model | Optimal shrinkage constant | Classical Markowitz problem | Portfolio Performance comparison |

(Briner and Connor 2008) | Demeaned SCVm | Market Model | Same as (Ledoit and Wolf 2004b) | N.A. | Analysis of the trade-off estimation error and model specification error |

(Pantaleo et al. 2011) | SCVm | Market Model, Common Covariance and Constant Correlation Model | Unbiased estimator of (Schäfer and Strimmer 2005) | Classical Markowitz problem | Portfolio Performance comparison |

(Candelon et al. 2012) | SCVm | Market Model and Identity | Same as (Ledoit and Wolf 2003, 2004b) | Black-Litterman GMVP | Portfolio Performance comparison |

(De Miguel et al. 2013) | SCVm | Scaled Identity | Expected quadratic loss and bootstrapping approach | Classical Markowitz problem | Comprehensive investigation of shrinkage estimators |

(Ardia et al. 2017) | SCVm | Market Model | Same as (Ledoit and Wolf 2003) | Risk-based portfolios | Theoretical paper to assess effect on risk-based weights |

Asset | Minimum Variance (MV) | Inverse Volatility (IV) | Equal-Risk-Contribution (ERC) | Maximum Diversification (MD) |
---|---|---|---|---|

Asset-1 | 0.500 | 0.400 | 0.448 | 0.506 |

Asset-2 | 0.500 | 0.400 | 0.374 | 0.385 |

Asset-3 | 0.000 | 0.200 | 0.177 | 0.108 |

**Table 3.**Frobenius norm for the portfolio weights. Values are averaged along with the shrinkage intensity (excluding the case $\delta =0$ ). For each $n$, the first line reports the Frobenius norm for the sample covariance matrix. Abbreviations in use are: S for sample covariance; Id for identity matrix; VId for Variance Identity; SI for Single-Index; CV for Common Covariance; CC for Constant Correlation and EWMA for Exponentially Weighted Moving Average.

P = 10 | P = 50 | P = 100 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MV | IV | ERC | MD | MV | IV | ERC | MD | MV | IV | ERC | MD | |

Panel A: n = 60 | ||||||||||||

S | 0.834 | 0.1585 | 0.1736 | 0.5842 | 0.7721 | 0.0573 | 0.0637 | 0.4933 | 0.7555 | 0.0409 | 0.0447 | 0.4565 |

Id | 0.6863 | 0.1425 | 0.1528 | 0.5045 | 0.6215 | 0.0559 | 0.0631 | 0.3873 | 0.4967 | 0.0404 | 0.0451 | 0.3652 |

VId | 0.6935 | 0.1583 | 0.1732 | 0.5176 | 0.5999 | 0.0567 | 0.0634 | 0.4092 | 0.5901 | 0.0404 | 0.0445 | 0.3686 |

SI | 0.838 | 0.1585 | 0.1736 | 0.5678 | 0.7685 | 0.0573 | 0.0637 | 0.4709 | 0.75 | 0.0409 | 0.0447 | 0.4288 |

CV | 1.2438 | 0.1583 | 0.1731 | 1.011 | 1.1484 | 0.0567 | 0.0628 | 0.9381 | 1.1386 | 0.0404 | 0.0438 | 0.9185 |

CC | 0.8353 | 0.1585 | 0.1733 | 0.5361 | 0.7808 | 0.0573 | 0.0635 | 0.4328 | 0.7663 | 0.0409 | 0.0445 | 0.3922 |

EWMA | 0.8473 | 0.1593 | 0.1745 | 0.595 | 0.7811 | 0.0575 | 0.064 | 0.5142 | 0.7325 | 0.0411 | 0.045 | 0.4431 |

Panel B: n = 120 | ||||||||||||

S | 0.9064 | 0.0877 | 0.0989 | 0.4649 | 0.7814 | 0.059 | 0.0656 | 0.5065 | 0.6519 | 0.0424 | 0.0472 | 0.4332 |

Id | 0.8157 | 0.087 | 0.0983 | 0.4256 | 0.6259 | 0.0613 | 0.0688 | 0.4354 | 0.6307 | 0.0389 | 0.0431 | 0.328 |

VId | 0.8235 | 0.0871 | 0.0985 | 0.4284 | 0.6259 | 0.0613 | 0.0688 | 0.4354 | 0.489 | 0.0421 | 0.0471 | 0.3712 |

SI | 0.9097 | 0.0877 | 0.0989 | 0.4563 | 0.7777 | 0.059 | 0.0656 | 0.4925 | 0.6458 | 0.0424 | 0.0472 | 0.419 |

CV | 1.3269 | 0.0871 | 0.0982 | 0.9667 | 1.1806 | 0.0587 | 0.0651 | 1.0138 | 1.0974 | 0.0421 | 0.0467 | 0.8951 |

CC | 0.905 | 0.0877 | 0.0988 | 0.4357 | 0.7822 | 0.059 | 0.0655 | 0.4636 | 0.6566 | 0.0424 | 0.0471 | 0.3856 |

EWMA | 0.9281 | 0.0883 | 0.0996 | 0.4859 | 0.7994 | 0.0592 | 0.0658 | 0.5246 | 0.6788 | 0.0427 | 0.0475 | 0.4601 |

Panel C: n = 180 | ||||||||||||

S | 0.7989 | 0.1311 | 0.1423 | 0.5007 | 0.7932 | 0.0564 | 0.0627 | 0.4631 | 0.6905 | 0.0404 | 0.044 | 0.4065 |

Id | 0.7206 | 0.1308 | 0.142 | 0.4736 | 0.6705 | 0.0562 | 0.0625 | 0.405 | 0.5477 | 0.0375 | 0.0399 | 0.3748 |

VId | 0.7273 | 0.1308 | 0.1421 | 0.4757 | 0.6838 | 0.0562 | 0.0626 | 0.4127 | 0.5754 | 0.0402 | 0.044 | 0.3556 |

SI | 0.8001 | 0.1311 | 0.1423 | 0.4954 | 0.7904 | 0.0564 | 0.0627 | 0.4545 | 0.6873 | 0.0404 | 0.044 | 0.3982 |

CV | 1.2715 | 0.1308 | 0.1419 | 0.9961 | 1.2073 | 0.0562 | 0.0624 | 0.9988 | 1.1422 | 0.0402 | 0.0437 | 0.8705 |

CC | 0.7957 | 0.1311 | 0.1423 | 0.4803 | 0.792 | 0.0564 | 0.0626 | 0.4259 | 0.692 | 0.0404 | 0.044 | 0.3672 |

EWMA | 0.8415 | 0.1322 | 0.1435 | 0.526 | 0.8284 | 0.0567 | 0.0631 | 0.5005 | 0.7206 | 0.0408 | 0.0445 | 0.4429 |

Panel D: n = 3000 | ||||||||||||

S | 0.7504 | 0.1476 | 0.1596 | 0.3957 | 0.734 | 0.049 | 0.0539 | 0.3988 | 0.513 | 0.0384 | 0.0428 | 0.3259 |

Id | 0.7441 | 0.1477 | 0.1597 | 0.3946 | 0.7009 | 0.049 | 0.0539 | 0.3872 | 0.4615 | 0.0384 | 0.0428 | 0.3096 |

VId | 0.7437 | 0.1477 | 0.1596 | 0.3945 | 0.7043 | 0.049 | 0.0539 | 0.3886 | 0.4673 | 0.0384 | 0.0428 | 0.312 |

SI | 0.7516 | 0.1476 | 0.1596 | 0.3955 | 0.7339 | 0.049 | 0.0539 | 0.3984 | 0.5123 | 0.0384 | 0.0428 | 0.3252 |

CV | 1.2864 | 0.1477 | 0.1597 | 0.963 | 1.2281 | 0.049 | 0.0538 | 0.9954 | 1.1041 | 0.0384 | 0.0428 | 0.6822 |

CC | 0.7488 | 0.1476 | 0.1596 | 0.3949 | 0.7316 | 0.049 | 0.0539 | 0.3904 | 0.5096 | 0.0384 | 0.0428 | 0.3143 |

EWMA | 0.8563 | 0.1489 | 0.1611 | 0.4452 | 0.8161 | 0.0497 | 0.0547 | 0.4652 | 0.6244 | 0.0389 | 0.0435 | 0.4076 |

Panel E: n = 6000 | ||||||||||||

S | 0.9672 | 0.1302 | 0.1409 | 0.4821 | 0.5737 | 0.0539 | 0.0589 | 0.3481 | 0.5772 | 0.0402 | 0.0437 | 0.3436 |

Id | 0.9496 | 0.1301 | 0.1408 | 0.4813 | 0.6095 | 0.0575 | 0.0639 | 0.4076 | 0.5449 | 0.0402 | 0.0437 | 0.3342 |

VId | 0.951 | 0.1301 | 0.1409 | 0.4815 | 0.5419 | 0.054 | 0.0589 | 0.3401 | 0.5483 | 0.0402 | 0.0437 | 0.3354 |

SI | 0.9688 | 0.1302 | 0.1409 | 0.482 | 0.574 | 0.0539 | 0.0589 | 0.3479 | 0.5772 | 0.0402 | 0.0437 | 0.3434 |

CV | 1.4142 | 0.1301 | 0.1408 | 1.0034 | 1.1436 | 0.054 | 0.0589 | 0.9706 | 1.1422 | 0.0402 | 0.0437 | 0.7031 |

CC | 0.9656 | 0.1302 | 0.1409 | 0.4814 | 0.5709 | 0.0539 | 0.0589 | 0.3415 | 0.575 | 0.0402 | 0.0437 | 0.3368 |

EWMA | 1.0432 | 0.1312 | 0.1422 | 0.5232 | 0.6946 | 0.0547 | 0.0599 | 0.4319 | 0.681 | 0.0407 | 0.0444 | 0.4229 |

**Table 4.**Frobenius norm for the portfolio weights. Values corresponds to the optimal shrinkage intensity, listed after the Frobenius norm for each portfolio. We report values for the sample covariance matrix ($\delta =0$) separately in the first row of each panel. For each $n$, the first line reports the Frobenius norm for the sample covariance matrix. Abbreviations used are: S for sample covariance; Id for identity matrix; VId for Variance Identity; SI for Single-Index; CV for Common Covariance; CC for Constant Correlation and EWMA for Exponentially Weighted Moving Average.

P = 10 | P = 50 | P = 100 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

MV | IV | ERC | MD | MV | IV | ERC | MD | MV | IV | ERC | MD | |

Panel A: n = 60 | ||||||||||||

S | 0.8340 | 0.1585 | 0.1736 | 0.5842 | 0.7721 | 0.0573 | 0.0637 | 0.4933 | 0.7555 | 0.0409 | 0.0447 | 0.4565 |

Id | 0.6778 | 0.1424 | 0.1525 | 0.501 | 0.5997 | 0.0558 | 0.0624 | 0.3704 | 0.471 | 0.0403 | 0.0446 | 0.3462 |

VId | 0.6689 | 0.1581 | 0.173 | 0.5084 | 0.5539 | 0.0565 | 0.0627 | 0.3795 | 0.5428 | 0.0402 | 0.0437 | 0.3331 |

SI | 0.8345 | 0.1585 | 0.1735 | 0.558 | 0.7666 | 0.0573 | 0.0637 | 0.4633 | 0.7479 | 0.0409 | 0.0447 | 0.4195 |

CV | 1.2392 | 0.1581 | 0.1729 | 0.509 | 1.117 | 0.0565 | 0.0627 | 0.3795 | 1.1068 | 0.0402 | 0.0437 | 0.3331 |

CC | 0.8335 | 0.1585 | 0.1731 | 0.5081 | 0.7733 | 0.0573 | 0.0634 | 0.3795 | 0.757 | 0.0409 | 0.0444 | 0.3332 |

EWMA | 0.8331 | 0.1586 | 0.1737 | 0.5852 | 0.7706 | 0.0573 | 0.0637 | 0.4953 | 0.7213 | 0.0409 | 0.0447 | 0.4395 |

Panel B: n = 120 | ||||||||||||

S | 0.9064 | 0.0877 | 0.0989 | 0.4649 | 0.7814 | 0.059 | 0.0656 | 0.5065 | 0.6519 | 0.0424 | 0.0472 | 0.4332 |

Id | 0.8121 | 0.087 | 0.0981 | 0.4241 | 0.6119 | 0.0613 | 0.0685 | 0.4255 | 0.613 | 0.0388 | 0.0428 | 0.3111 |

VId | 0.8121 | 0.087 | 0.0982 | 0.4242 | 0.6119 | 0.0613 | 0.0685 | 0.4255 | 0.4425 | 0.042 | 0.0467 | 0.3445 |

SI | 0.907 | 0.0877 | 0.0989 | 0.4526 | 0.776 | 0.059 | 0.0656 | 0.4872 | 0.6431 | 0.0424 | 0.0472 | 0.414 |

CV | 1.3269 | 0.087 | 0.0981 | 0.4245 | 1.1756 | 0.0586 | 0.0651 | 0.4302 | 1.0916 | 0.042 | 0.0467 | 0.3445 |

CC | 0.9043 | 0.0877 | 0.0987 | 0.4241 | 0.781 | 0.059 | 0.0654 | 0.4302 | 0.6527 | 0.0424 | 0.0471 | 0.3446 |

EWMA | 0.9052 | 0.0876 | 0.0988 | 0.4651 | 0.7797 | 0.0589 | 0.0655 | 0.5056 | 0.6554 | 0.0424 | 0.0472 | 0.4331 |

Panel C: n = 180 | ||||||||||||

S | 0.7989 | 0.1311 | 0.1423 | 0.5007 | 0.7932 | 0.0564 | 0.0627 | 0.4631 | 0.6905 | 0.0404 | 0.044 | 0.4065 |

Id | 0.7177 | 0.1307 | 0.1419 | 0.4724 | 0.6613 | 0.0562 | 0.0624 | 0.3977 | 0.534 | 0.0375 | 0.0398 | 0.3645 |

VId | 0.718 | 0.1307 | 0.1419 | 0.4724 | 0.6614 | 0.0562 | 0.0624 | 0.3979 | 0.5428 | 0.0402 | 0.0437 | 0.3331 |

SI | 0.799 | 0.1311 | 0.1423 | 0.4929 | 0.7897 | 0.0564 | 0.0627 | 0.4515 | 0.6863 | 0.0404 | 0.044 | 0.3955 |

CV | 1.2715 | 0.1307 | 0.1418 | 0.4724 | 1.2073 | 0.0562 | 0.0624 | 0.3979 | 1.1422 | 0.0402 | 0.0437 | 0.3331 |

CC | 0.7942 | 0.1311 | 0.1422 | 0.4725 | 0.7912 | 0.0564 | 0.0626 | 0.3977 | 0.6904 | 0.0404 | 0.0439 | 0.3331 |

EWMA | 0.8035 | 0.1312 | 0.1424 | 0.5008 | 0.7951 | 0.0564 | 0.0626 | 0.4653 | 0.6938 | 0.0404 | 0.044 | 0.4074 |

Panel D: n = 3000 | ||||||||||||

S | 0.7504 | 0.1476 | 0.1596 | 0.3957 | 0.734 | 0.049 | 0.0539 | 0.3988 | 0.513 | 0.0384 | 0.0428 | 0.3259 |

Id | 0.7425 | 0.1477 | 0.1596 | 0.3941 | 0.6988 | 0.049 | 0.0538 | 0.3859 | 0.4573 | 0.0384 | 0.0428 | 0.3072 |

VId | 0.7426 | 0.1476 | 0.1596 | 0.3941 | 0.6988 | 0.049 | 0.0538 | 0.3859 | 0.4573 | 0.0384 | 0.0428 | 0.3072 |

SI | 0.7506 | 0.1476 | 0.1596 | 0.3953 | 0.7339 | 0.049 | 0.0539 | 0.3983 | 0.512 | 0.0384 | 0.0428 | 0.325 |

CV | 1.2864 | 0.1476 | 0.1596 | 0.3951 | 1.2281 | 0.049 | 0.0538 | 0.3859 | 1.1041 | 0.0384 | 0.0428 | 0.3072 |

CC | 0.7477 | 0.1476 | 0.1596 | 0.3946 | 0.7299 | 0.049 | 0.0539 | 0.386 | 0.5073 | 0.0384 | 0.0428 | 0.3072 |

EWMA | 0.7615 | 0.1477 | 0.1597 | 0.3981 | 0.7439 | 0.0491 | 0.0539 | 0.4043 | 0.5263 | 0.0384 | 0.0429 | 0.3346 |

Panel E: n = 6000 | ||||||||||||

S | 0.9672 | 0.1302 | 0.1409 | 0.4821 | 0.5737 | 0.0539 | 0.0589 | 0.3481 | 0.5772 | 0.0402 | 0.0437 | 0.3436 |

Id | 0.9486 | 0.13 | 0.1408 | 0.4811 | 0.6085 | 0.0575 | 0.0639 | 0.4072 | 0.5428 | 0.0402 | 0.0437 | 0.3331 |

VId | 0.9486 | 0.13 | 0.1408 | 0.4811 | 0.5365 | 0.054 | 0.0589 | 0.3381 | 0.5428 | 0.0402 | 0.0437 | 0.3331 |

SI | 0.9675 | 0.1302 | 0.1409 | 0.482 | 0.5738 | 0.0539 | 0.0589 | 0.3478 | 0.5772 | 0.0402 | 0.0437 | 0.3433 |

CV | 1.4142 | 0.13 | 0.1408 | 0.4811 | 1.1436 | 0.054 | 0.0589 | 0.3381 | 1.1422 | 0.0402 | 0.0437 | 0.3331 |

CC | 0.9644 | 0.1302 | 0.1409 | 0.4812 | 0.5687 | 0.0539 | 0.0589 | 0.3381 | 0.5733 | 0.0402 | 0.0437 | 0.3331 |

EWMA | 0.9765 | 0.1302 | 0.1409 | 0.4832 | 0.5901 | 0.054 | 0.059 | 0.3561 | 0.59 | 0.0402 | 0.0438 | 0.3524 |

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Neffelli, M.
Target Matrix Estimators in Risk-Based Portfolios. *Risks* **2018**, *6*, 125.
https://doi.org/10.3390/risks6040125

**AMA Style**

Neffelli M.
Target Matrix Estimators in Risk-Based Portfolios. *Risks*. 2018; 6(4):125.
https://doi.org/10.3390/risks6040125

**Chicago/Turabian Style**

Neffelli, Marco.
2018. "Target Matrix Estimators in Risk-Based Portfolios" *Risks* 6, no. 4: 125.
https://doi.org/10.3390/risks6040125