You will be provided with a reference and some statements. Please determine whether each statement is 'supported', 'unsupported', or 'unknown' with respect to the reference. Please note:
First, assess whether the reference contains any valid content. If the reference contains no valid information, such as a 'page not found' message, then all statements should be considered 'unknown'.
If the reference is valid, for a given statement: if the facts or data it contains can be found entirely or partially within the reference, it is considered 'supported' (data accepts rounding); if all facts and data in the statement cannot be found in the reference, it is considered 'unsupported'.

You should return the result in a JSON list format, where each item in the list contains the statement's index and the judgment result, for example:
[
    {
        "idx": 1,
        "result": "supported"
    },
    {
        "idx": 2,
        "result": "unsupported"
    }
]

Below are the reference and statements:
<reference>
Mean-variance and hierarchical risk parity:
An empirical study of large-cap stock
portfolios
Guanqi Zhu, MAFN masters degree candidate, Department of
Mathematics, Columbia University.
Jonathan Schachter, Ph.D., CEO, Delta Vega Inc.

Abstract: This study compares two portfolio strategies: Markowitz Mean
Variance (MV) and Hierarchical Risk Parity (HRP). We investigate how
HRP's robust, diversification-focused approach performs against MV's
theoretically optimal but often unstable method, using real-world data
from 15 large-cap stocks.

Introduction & Problem
●​ Markowitz (MV) Model: As the cornerstone of modern portfolio theory,
the MV model aims to maximize risk-adjusted returns by balancing
expected return and variance. However, in practice, it faces two major
challenges:
○​ Data Instability: The model relies on precise estimates of future
returns and a stable covariance matrix, which are often uncertain
and noisy.
○​ Matrix Pathology: With limited data samples, the covariance
matrix can become ill-conditioned, a state that amplifies
estimation errors and leads to extreme, unstable weight
allocations.
●​ Hierarchical Risk Parity (HRP) Model: HRP is a machine learning
algorithm that uses hierarchical clustering. Developed as a robust
alternative, HRP bypasses the need for matrix inversion by using this
clustering method to identify the correlation structure between assets. It
then allocates risk recursively to achieve a more balanced and stable
portfolio.

1

HRP Method: Correlation Clustering
The core of the HRP model is to identify the correlation structure of assets
through hierarchical clustering, thereby building a more robust portfolio and
avoiding the need for covariance matrix inversion.
Original Matrix: Displays the unprocessed correlation matrix of assets. The
correlations are dispersed and lack a clear structure.

Clustered Matrix: Shows the reordered correlation matrix after applying the
HRP algorithm, where highly correlated assets are grouped together to form
distinct "blocks."

2

Weight Allocation: Concentration vs. Diversification
The fundamental difference between the MV and HRP models lies in their
weight allocation strategies. The MV model tends to concentrate on a few
assets, while the HRP model is dedicated to a broader diversification of risk.
●​ MV's Concentration: A few stocks, such as MA, receive extremely high
weights in the MV model (points are on the right side of the chart),
which reflects the model's concentrated allocation in pursuit of
maximizing returns.
●​ HRP's Breadth: Stocks like UL, CVS, and TM receive higher weights in
the HRP model (points are in the upper part of the chart), indicating that
HRP spreads risk across a broader range of assets.
●​ Key Insight: HRP's allocation strategy allows for a wider diversification
of risk, whereas MV may incur higher concentration risk in the pursuit of
high returns.

3

Robustness Analysis: Marginal Impact
●​ Sharpe Ratio Change: The removal of stocks like TGT and DIS
significantly increased the Sharpe ratio for both models, suggesting
these stocks may have been a drag on the original portfolio's
risk-adjusted returns.

●​ Volatility Change: When WMT was removed, the MV portfolio's volatility
rose dramatically, while the HRP portfolio's volatility remained largely
unchanged. This demonstrates HRP's robustness.

4

Return Distribution Comparison: MV vs HRP
The Q-Q plot provides a direct visual comparison of the monthly return
distributions for the MV and HRP models, revealing their performance
differences in both typical and extreme market scenarios.
●​ Central Performance: In most return scenarios, the two models'
performance is highly similar, with the data points clustering tightly
around the baseline.
●​ Tail Performance: In extreme return scenarios, HRP's losses are
smaller than MV's (less downside risk), but its gains are also smaller
than MV's (limited upside potential).
●​ Key Insight: HRP's risk parity strategy effectively limits downside risk,
but at the cost of sacrificing some upside gains.

Skew

Excess Kurtosis

Jarque–Bera P Value

MV

-0.1957

0.0419

0.730

HRP

0.0140

0.2576

0.800

5

Core Conclusion and Trade-off
The empirical analysis in this study reveals a key trade-off between the
Markowitz (MV) and Hierarchical Risk Parity (HRP) models: Efficiency versus
Robustness.
●​ MV's Efficiency: In a favorable data environment (e.g., with a low
covariance matrix condition number: the ratio of the largest to smallest
eigenvalues), the MV model can effectively leverage its theoretical
advantages to achieve higher average returns and superior downside
risk-adjusted returns (a higher Sortino ratio).
●​ HRP's Robustness: Although the HRP model is more conservative in
terms of returns, its risk diversification strategy, achieved through
hierarchical clustering, makes it more robust to changes in asset
composition. Its return distribution is closer to normal, and it is better at
limiting extreme losses.
The Core Trade-off: Choosing the MV model means pursuing higher return
efficiency but accepting higher concentration risk and greater sensitivity to
data uncertainty. Conversely, choosing the HRP model means accepting more
moderate returns in exchange for greater robustness and more predictable
risk characteristics. The ultimate choice depends on the investment manager's
core objective: whether to pursue efficiency or stability.

6

References
Black, F., & Litterman, R. (1992). Global Asset Allocation with Equities, Bonds, and
Currencies. Financial Analysts Journal, 48(5), 28–43.
Cochrane, J. H. (2005). Asset Pricing. Princeton University Press.

Fama, E. F. (1970). Efficient Capital Markets: A Review of Theory and Empirical Work.
The Journal of Finance, 25(2), 383–417.
Grinold, R. C., & Kahn, R. N. (2000). Active Portfolio Management. McGraw-Hill
Education.
Lopez de Prado, M. (2020). Machine Learning for Asset Managers. Cambridge University
Press.
Maillard, S., Roncalli, T., & Teïletche, J. (2010). The Properties of Equally Weighted Risk
Contribution Portfolios. Journal of Portfolio Management, 36(4), 103–113.
Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91.
Meucci, A. (2010). The Black-Litterman Approach: Original Model and Extensions. In
Risk and Asset Allocation. Springer.
Sharpe, W. F. (1966). Mutual Fund Performance. The Journal of Business, 39(1), 119–138.

7
</reference>

<statements>
1. The foundational assumption of Markowitz optimization is that asset returns follow a multivariate normal (or elliptical) distribution fully described by its first two central moments, with the parameters \(\mu\) and \(\Sigma\) assumed to be known with deterministic certainty
2. In practical regimes where the historical sample length \(T\) is of comparable magnitude to the universe size \(N\), the sample covariance matrix \(\hat{\Sigma}\) becomes ill-conditioned, exhibiting an elevated condition number
3. Furthermore, the assumption of multivariate normality fails during market dislocations, as empirical financial returns exhibit pronounced negative skewness, excess kurtosis, and volatility clustering
4. Because asset prices exhibit weak serial correlation and pronounced non-stationarity, historical arithmetic means are notoriously poor predictors of future drift
5. More critically, it assumes that portfolio risk can be modeled without considering higher-order moments, leaving portfolios exposed to tail-risk events
6. Second, graph-theoretic and unsupervised clustering algorithms, such as Hierarchical Risk Parity (HRP), calculate risk over hierarchical correlation distance metrics, avoiding the numerical instability of full matrix inversion
7. In the Markowitz Mean-Variance (MVO) framework, Tail Risk & Non-Linearity fails to account for fat tails, skewness, or correlation breakdowns
8. To mitigate the instability of inverting empirical covariance matrices, Marcos López de Prado introduced Hierarchical Risk Parity (HRP)
9. By combining unsupervised machine learning (hierarchical clustering) with graph theory, HRP decomposes portfolio construction into three distinct stages
10. First, tree clustering groups assets into a hierarchical dendrogram based on their pairwise correlation distance metric \(D_{i,j} = \sqrt{\frac{1}{2}(1 - \rho_{i,j})}\)
11. Second, quasi-diagonalization reorganizes the covariance matrix so that highly correlated assets are grouped contiguously along the main diagonal
12. Third, recursive bisection allocates capital top-down across dendrogram branches based on cluster inverse-variance, descending to individual assets without requiring a full matrix inversion
13. Empirical evaluations confirm that HRP generates stable, diversified portfolio allocations that limit drawdown risk during market dislocations while avoiding the extreme concentration typical of ill-conditioned optimizers
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.