Connectedness optimization across four assets returns equal weight

Minimizing connectedness across four weakly dependent markets produces something close to equal weight. Bouzguenda and Jarboui's table gives gold 0.250, WTI 0.249, S&P 500 0.235 and SSE 0.266. Those weights are recalculated with every daily observation.

The paper proposes five allocation rules for managing systemic risk across energy and financial markets through the COVID-19 pandemic and the Russia-Ukraine conflict. Its claimed contribution rests on resilience during crises. The method used to build the portfolios is more interesting than the resulting allocations.

How covariance becomes a spillover network

Bouzguenda and Jarboui use daily closing prices for gold, WTI crude, the S&P 500 and the Shanghai Composite from 3 January 2019 to 1 August 2025. Prices are in USD, and returns are continuously compounded log-returns. Each series is fitted with a univariate GARCH(1,1) from the Hentschel flexible family. The specification is selected through the Antonakakis et al. adjusted-BIC criterion, followed by knock-out gates based on misspecification tests. Pair-specific bivariate DCC recursions supply the time-varying conditional covariance matrix.

The distinctive step comes after that. The authors substitute the conditional matrix into multivariate regression algebra to derive dynamic conditional betas and a dynamic conditional R-squared. On each date, the latter measures how much of one market's conditional variance is explained by the other three. Genizi's PCA-based decomposition assigns that R-squared across individual markets, solving the attribution problem created when the regressors are correlated. The resulting decomposed matrix replaces the forecast-error variance decomposition inside the Diebold-Yılmaz framework. From it come TO, FROM, NET and the total connectedness index.

The portfolio rules follow directly from those quantities. A market currently absorbing variance from the other assets could be underweighted before a covariance-only optimizer reacts. The paper compares five long-only, fully invested portfolios: minimum variance using the covariance matrix, minimum correlation, minimum connectedness using the pairwise connectedness index, minimum bivariate R-squared, and minimum decomposed-R-squared connectedness. For the Kroner-Ng bivariate weights, the authors explicitly impose a 0-to-1 no-short constraint.

Average total connectedness is 24.41%, which the paper interprets as roughly a quarter of each market's forecast error variance originating elsewhere. The S&P 500 transmits 9.47% and receives 9.28%. Its net position, the system's largest, is +0.20, and it records a net-transmitter count of 3 out of 3. Gold sends 6.17% and receives 6.24%, while WTI sends 6.38% and receives 6.45%. The SSE sends 2.39% and receives 2.44%. Apart from the S&P, every net figure falls between -0.05 and -0.07 percentage points. Those are rounding-scale imbalances in an almost symmetric matrix. Kendall taus range from 0.013 for gold against the S&P, which is insignificant, to 0.122 for WTI against the S&P.

The paper gives conflicting accounts of the main transmitter. Its abstract calls WTI "a principal conduit of volatility spillovers"; the conclusion says that "the S&P 500 is identified as the main channel for shock transmission." The averaged table supports the conclusion. WTI is a mild net receiver at -0.07.

Almost equal weight, recalculated daily

The four minimum connectedness weights range from 0.235 to 0.266. For the decomposed version, the range is 0.242 to 0.259. Their cross-time dispersions are 0.012 to 0.017 and 0.007 to 0.009 respectively.

Minimum variance looks quite different. It allocates 0.347 to gold, 0.018 to WTI, 0.379 to the S&P and 0.256 to the SSE. Weight dispersion runs from 0.024 on the 1.8% WTI sleeve to 0.202 on the S&P. In a four-asset network with such weak links, the connectedness objective is close to non-binding. The position amounts to 1/N plus a daily DCC re-estimation.

The scorecard reflects that similarity. The paper reports three Sharpe rows, and none is led by the connectedness family. Using standard deviation, minimum variance ranks first at 0.6242. Minimum correlation follows at 0.5099, against 0.4397 for minimum connectedness, 0.4393 for minimum bivariate R-squared and 0.4494 for the decomposed version. On the VaR-based ratio, minimum correlation leads at 6.9254, followed by minimum variance at 5.9791 and the decomposed portfolio at 5.5132. Minimum correlation also leads on CVaR at 6.9254. Minimum variance is genuinely worst there at 3.2311, while the connectedness portfolios score 5.3912 to 5.5132 and beat it. This is their sole win. The three connectedness results fit inside a 0.010 band: 0.4397, 0.4393 and 0.4494. We found no test comparing their Sharpe differences.

Those levels require care. The ratios use daily returns, without annualization or a risk-free rate. Minimum variance has a reported dispersion of 0.0008841. Annualized, that would be about 1.4% volatility for a portfolio holding 34.7% gold and 37.9% S&P 500. Whatever the denominator represents, the result cannot be compared with an annualized Sharpe. In the paper's performance table, VaR and CVaR ratios across the five portfolios span 3.2 to 6.9, which is another statistic.

The authors acknowledge the limits of hedging effectiveness. "A high HE indicates effective risk reduction, but it does not guarantee improved risk-adjusted returns," the paper says. Its results make the problem plain. The WTI/SSE optimal-weight pair records hedging effectiveness of 0.900 and a Sharpe of -0.154. SP500/WTI produces HE of -0.035 and a Sharpe of 0.209. Across all five minimum-risk portfolios, WTI has HE between 0.899 and 0.968. That sits awkwardly with the paper's account of gold as the stabilizer. Gold's HE is negative in four portfolios, ranging from -0.119 to -0.287.

Where is the crisis evidence?

The abstract concedes the full-sample ranking in one sentence: "While the Minimum Variance Portfolio delivers robust risk-adjusted returns, strategies based on connectedness metrics demonstrate superior resilience during crises." Its second clause carries the paper's main claim. The conclusion goes further, saying that adaptive strategies which respond to changing asset interdependencies "outperform traditional approaches based on variance-covariance or correlation matrices, particularly in terms of hedging effectiveness and risk-adjusted returns during crises."

The full-sample performance table runs the other way. Minimum variance scores 0.6242 and minimum correlation 0.5099, both above the connectedness family's 0.4393 to 0.4494.

Evidence for resilience comes from the path of connectedness rather than portfolio performance within the crises. The TCI spikes in early 2020 and again in 2022. DCC correlations between the S&P and WTI or gold exceed 0.4 in 2020 and 2022-2023, whereas the SSE rarely exceeds 0.2. A crisis-subsample performance table covering the five portfolios does not appear; the published performance table covers the full sample. The superiority claim therefore moves from a diagnostic result to an allocation conclusion. One awkward detail remains: the paper describes a TCI spike "around 2018", before its stated sample begins.

We also could not find a description of rolling or out-of-sample re-estimation. If the DCC parameters are estimated once over 2019-2025, every historical weight incorporates information from the entire period. A live implementation would face less flattering risk numbers. Transaction costs, turnover and rebalancing frictions are also unreported, although the weights change daily.

Our run used ETF proxies

Spot gold, WTI log-price returns and the SSE Composite are unavailable to us as tradable instruments. We therefore adapted the decomposed-connectedness allocation to GLD, USO, SPY and FXI. This tests the mechanism through substitute exposures because USO has roll mechanics and FXI represents a different China-equity composition.

We fitted pair-specific bivariate DCC models on a 252-session trailing window and traded at the next session's close. The portfolio remained long-only and fully invested. We also penalized net transmitters. NET is each market's transmitted-minus-received value in the connectedness table. Placing max(NET, 0) on the diagonal shifts weight away from markets sending more shocks than they receive. The paper does not apply this penalty.

Our figures cover 2 January 2020 to 1 July 2024: Sharpe 0.205, CAGR 2.39%, total return 11.20%, volatility 12.16%, maximum drawdown -24.8%, Calmar 0.10, 2,309 trades. The paper gives 0.4494 for its decomposed-connectedness portfolio and 0.4397 for minimum connectedness, both above our 0.205. Its figures divide a raw daily mean by dispersion of about 0.0016 a day, or roughly 2.5% annualized. Our Sharpe is annualized using 12.16% realized volatility. The universe and window also differ.

Beta to SPY was 0.19. The transmitter penalty reduces exposure to the S&P 500, the period's strongest performer, and reallocates toward net receivers. During 2021-2024, that meant FXI. USO also suffered from the 2020 contango destruction. Our window starts with the COVID crash and ends before the 2024-2025 gold advance. It includes 2,309 daily rebalances under a per-share commission with a one dollar minimum, and we cannot confirm whether the 11.20% result is gross or net of that charge.

Each of these choices pushes in the same direction. We cannot measure their contribution relative to the paper's figures, so we do not claim that they explain the gap. This was one automated pass, and its evidence bears first on our implementation rather than the authors' work.

A crisis-window table would change my view. It should report returns and drawdowns for the five portfolios over, say, February to June 2020 and February to June 2022, using rolling out-of-sample DCC estimates and deducting the cost of daily rebalancing. Until then, decomposed R-squared connectedness offers a useful account of which market absorbs whose variance. The portfolio derived from it remains 1/N with extra steps.

Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.

For each session t after the close:
  1. Compute demeaned close-to-close log returns for GLD, USO, SPY, and FXI.
  2. Require 252 complete trailing observations; otherwise retain the last valid target.
  3. Fit each Hentschel-class GARCH(1,1), selecting its innovation distribution by adjusted BIC and diagnostic gates.
  4. Fit pair-specific bivariate DCC recursions and construct conditional H_t and R_t.
  5. Compute conditional R-squared values and their Genizi directional decomposition.
  6. Calculate FROM_i, TO_i, and NET_i = TO_i - FROM_i.
  7. Construct the Equation-27 connectedness matrix PCI_C,t.
  8. Set P_t = PCI_C,t + diag(max(NET_t, 0)).
  9. Solve min_w w'P_t w subject to sum(w)=1 and 0 <= w_i <= 1.
 10. If estimation, decomposition, or optimization fails, retain the previous valid holdings.
 11. Submit the target rebalance for the next valid session's close; apply it only after execution.