Research on volatility and correlation of dry bulk market

Studies examining the fluctuations in dry bulk markets often fall into two distinct categories. The first focuses on the volatility and correlation between the dry bulk market and other markets, while the second investigates the volatility and correlation within the dry bulk market itself. For instance, [2] utilized non-parametric causal quantiles to analyze the asymmetric relationship between the BDI and the spot prices of bulk commodities, finding that fluctuations in commodity prices can lead to spillovers in BDI volatility. However, the impact of the BDI on commodity prices varies significantly depending on the type of commodity and prevailing market conditions. For example, [3] identified a notable correlation between the BDI and crude oil prices, observing that this relationship is strong in the short term but weakens over the long term. Similarly, [4] employed Granger causality tests and co-integration analysis to explore the leading-lag relationships among the BDI, the Shanghai Container Freight Index (SCFI), and the Baltic Dirty Tanker Index (BDTI). They also applied multivariate impulse response functions and variance decomposition to assess how the freight market reacts to shocks in other freight markets. Their findings indicate that the dry bulk market is influenced by fluctuations in both the container and tanker shipping markets, with mutual volatility conduction observed specifically between the dry bulk and container shipping markets.

There is a substantial body of literature examining the volatility correlations within the primary dry bulk market. For example, [5] employed a hybrid model combining wavelet analysis and neural networks to investigate the fluctuations of freight indices for the 2A and 3A routes in the BPI market. Their time-series wavelet multi-scale decomposition highlighted the dynamics of fluctuations across different time frequencies. Similarly, [6] utilized the multifractal detrending volatility analysis (MD-DFA) technique to analyze market trends in the dry bulk freight indices of Capesize and Panamax vessels. Furthermore, [7] demonstrated that the contribution of the BPI market to the BDI has gradually increased, underscoring the significance of the Panamax bulk carrier market and its considerable influence on the development of the BDI. In another study, [8] applied Rescaled Range Analysis (R/S) and an enhanced R/S analysis method to examine the long-term memory of two shipping submarkets based on the freight indices of Panamax and Handysize vessels. While there have been numerous achievements regarding the fluctuation correlations between dry bulk markets and external factors, as well as within the internal primary market, there remains a notable gap in research focusing on the volatility and correlation of freight rates across various routes within the internal dry bulk markets.

Research on risk spillovers of international dry bulk market

The sub-segments of the dry bulk shipping market are interconnected, and previous studies have highlighted the spillover effects among these sub-segments. There are differences in the response of dry bulk shipping market sub-segments to shocks [9]. There are some studies on risk spillovers between dry bulk market segments or with other markets. For instance, [10] was the first to combine long memory processes with Value at Risk (VaR) to examine risk spillover within the international dry bulk shipping market. Additionally, [11] utilized a three-variable VAR-BEKK-GARCH-X model to analyze the spillover effects between the BDI and the financial markets, and find that these spillover effects are time-varying and become more pronounced during the 2008–2009 global financial crisis. Furthermore, [12] employed a GARCH-Copula-CoVaR approach to investigate the extreme risk spillovers from the commodity market to the maritime sector, taking into account the interactions among different sub-sectors of the maritime market. Meanwhile, [13] explored risk measures, risk attitudes, and variable control related to the freight rate cycle in the dry bulk shipping market across various scenarios, concluding that there is a negative correlation between risk and return in long-term contracts. In addition, The dry bulk industry and the oil market often study their risk spillovers together. Compared with the dry bulk market, the oil tanker market has a higher integration degree, the high spillover period lasts longer, and the oil price fluctuation contributes more to the spillover effect of the oil tanker market [14]. Despite these contributions, there is a notable gap in the literature regarding risk spillover between routes within the dry bulk shipping market. Therefore, this paper aims to address this gap, providing valuable data support for the market planning of relevant shipping enterprises.

Research on empirical methodology

Compared to the ARCH model, the GARCH model offers a linear extension of variance representation, effectively addressing the computational inefficiencies and accuracy limitations associated with high-order ARCH models. Given the GARCH family’s capacity to fit marginal distributions, it is particularly suitable for analyzing the volatility of freight rates in the dry bulk market [15]. [16] introduced the Copula function, which posits that the joint distribution of multivariate variables can be expressed as a function that combines the marginal probability distributions of each variable with a description of the correlation structure among them. The Copula function has found widespread application across various fields ([16, 17]). VaR is the most commonly used method for measuring the risk of individual institutions. However, it may not adequately reflect systemic risk, especially during periods of market instability. To address this gap, [18] introduced tail correlation analysis and proposed the Conditional Value at Risk (CoVaR) method. The Copula function is employed to calculate CoVaR ([19–21]).

The hybrid GARCH-Copula-CoVaR method has gained significant traction in modeling volatility correlations and risk spillovers across financial markets. For instance, [22] employed various GARCH-Copula models to analyze the tail dependencies among oil prices, investor expectations, and stock returns. Similarly, [23] introduced a GARCH-Copula deformation model to assess whether gold, the US dollar, and Bitcoin serve as hedging or safe-haven assets for stocks, and their potential in diversifying downside risks in international stock markets. Recently, this model has been extended to other fields, such as energy; [24] utilized the Copula framework to uncover the nonlinear tail-dependent structures between carbon and energy markets, calculating CoVaR to quantify extreme risk spillover effects. Findings reveal that during extreme events, risk spillovers from both traditional and renewable energy markets to carbon markets significantly increase. The GARCH-copula regression model is also used to analyze the heterogeneity of dynamic risk spillovers between logistics market and e-commerce market [25]. In the maritime sector, the GARCH-Copula-CoVaR approach offered new insights into risk transmission from oil and energy markets to maritime markets, highlighting interactions among various subsectors within the maritime industry [26].

Unlike [26], which focused on the risk spillover effects between oil, the ex-energy sector, and the BDI market, this paper investigates the volatility, correlation, and risk spillover effects among the main routes of the dry bulk shipping sub-market. Specifically, when the freight rate for a particular route experiences a sharp rise or fall, the tail dependence structure between that route and others is non-linear. To explore this phenomenon, we calculate the CoVaR using a static GARCH-Copula approach to examine risk spillover between routes. Additionally, we employ time-varying copula functions to assess the dynamic and tail correlations between the BCI and the BPI markets. Given that declines in freight rates typically result in greater losses, this paper primarily focuses on risk spillover during periods of market decline.

Marginal distribution model

The GARCH model effectively captures the time-varying volatility of financial time series, reflecting market volatility characteristics across different periods, which is essential for understanding and predicting financial risk. To account for the effects of autocorrelation and positive and negative shocks on conditional fluctuations, researchers often integrate an ARMA(r,s) model with the GARCH(p,q) framework. In this paper, we utilize the ARMA(r,s)-GARCH(p,q) model to analyze the time series of freight rates and derive the edge distribution. (1) The first line of Eq (1) represents the mean value equation of the ARMA(r,s) model, while the third line denotes the conditional variance equation. Here, γ is the maximum lag order of the autoregressive term, which influences the complexity of the autoregressive component, and s is the maximum lag order of the moving average, determining its impact on the model. μ is the constant term, φ_i is the coefficient of AR(r), θ_j is the coefficient of MA(s), and ω is the mean of conditional variance regression term. a_t is the residual term. Since the freight rates collected in this paper has a heavier tail, ARMA(r,s)-GARCH(p,q) under the skew-t distribution is selected. σ_t denotes the conditional standard deviation, which measures return volatility, while indicates the conditional variance, capturing the squared effects of these fluctuations. The perturbation term {ϵ_t} is typically assumed to be an independent and identically distributed sequence of random variables with a mean of 0 and a variance of 1.

Static and dynamic Copula functions

The Copula method is able to model dependencies between multiple variables, especially in cases where the data does not conform to a normal distribution. It captures tail dependencies between financial assets, which is particularly important during periods of market turmoil. When the random variables with different edge distributions are not independent, the traditional joint distribution fitting method is inadequate. To solve this puzzle, the Copula function can be used to fit the joint distribution accurately and flexibly for random variables with multiple edge distributions.

Binary Sklar’s theorem: Let H(⋅, ⋅) be a joint distribution function with edge distributions F(⋅) and G(⋅), then there is a function C(⋅, ⋅) such that: (2) where C(⋅, ⋅) is corresponding Copula function. The original function H(x, y) can be written as a function C(⋅, ⋅) related on u and v, then the function C(⋅, ⋅) is the Copula function, that is, (3)

To characterize the properties of freight rates exhibiting peak and thick tails, we introduce six static copula functions to describe the nonlinear relationships and tail correlations. Additionally, we incorporate three time-varying copula functions to capture the dynamic changes in correlation between the freight rates in the BCI and BPI markets. The formulas for these copula functions are presented in Table 1.

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Table 1. Static and dynamic Copula functions.

https://doi.org/10.1371/journal.pone.0315167.t001

CoVaR methodology

1. 1. CoVaR. VaR is primarily used to assess the risk associated with a specific asset, but it fails to capture systemic risk within a market. Additionally, VaR is limited in its ability to measure risks that arise under extreme conditions. Therefore, this paper considers the CoVaR approach, which builds on the foundation of VaR. The CoVaR model measures the potential losses of one route when a risk materializes in another route, making it particularly valuable for analyzing financial system stability and assessing systemic risk. For example, the extreme loss of the freight return Y_s,t occurs under the condition that the loss is defined as follows: (4) CoVaR can be divided into ascending CoVaR and descending CoVaR according to different confidence levels. above is the descending condition at risk value of Y_s,t when Y_i,t is under the condition of descending risk. The corresponding ascending CoVaR expression is as follows: (5) where β^d + β^u = 1, β^u = 0.05 and β^d = 0.95.
2. 2. ΔCoVaR. Whether or they still included in the calculation of VaR, can not measure spillover effect between the two routes. Therefore, ΔCoVaR is usually used to measure the risk spillover effect between different freight rates of various routes. For example, the downside risk spillover of the freight return for s route under the freight return of i route is expressed as follows: (6) Correspondingly, the downside risk spillover of the freight rate for i route under the freight rate return of s route is as follows: (7)
3. 3. %CoVaR. Although ΔCoVaR has been calculated to measure the risk spillover effect, the %CoVaR is continued to be considered to remove the dimensional effect. For example, the %CoVaR of the freight rate return for s route to the freight rate return of i route is as follows: (8) %CoVaR represents a relative change, and the %CoVaR rank is used to compare the risk spillover degree between different routes. Since the extreme loss mainly occurs in the period of economic downturn, we mainly studies the risk spillover effect in the downward state, and so ΔCoVaR and %CoVaR are.

GARCH-Copula-CoVaR model

The hybrid Garch-Copula-CoVaR model used the GARCH model to describe the marginal distribution of the freight return of each route. Set the marginal distribution of the freight return for the routes as G(t) and the edge density function as g(t). c(⋅, ⋅) is obtained from the first derivative of the optimal Copula, that is, . The specific steps are as follows.

1. Step 1: (X, Y) are defined as two-dimensional continuous random variables. f(x) represents the edge distribution density function of X. g(y) represents the edge distribution density function of Y. h(X, Y) is defined as the joint density function of (X, Y). Then we have (9) From binary Sklar’s theorem, we can get (10)
2. Step 2: Substitute (10) into (4), then one can obatin (11)
3. Step 3: Then we achieve (12) Hence, can be calculated by inverse solution method. the and can be get by (7) and (8), respectively.

Data source

We selected the daily freight rates for the main routes in the BCI and BDI markets, which are widely utilized in maritime economics research. The BCI serves as an index for the Capesize dry bulk market and is constructed based on the spot freight rates of Capesize dry bulk carriers, known for their substantial capacity in the dry bulk shipping sector. In the transportation of dry bulk cargo, iron ore represents the largest share of total sea freight. The C2, C3 and C5 routes are the three most significant pathways for iron ore transport, making them ideal representatives of the BCI market.

The BPI is an index that represents the Panamax dry bulk market, based on the spot freight rates of Panamax dry bulk carriers. These vessels play a crucial role in the dry bulk shipping sector due to their significant capacity. The P1A route connects the Americas to Western Europe, P2A links the Far East with the Atlantic Ocean, and P3A connects the Americas to the Far East. These routes are among the busiest in the world. Given that P1A_03, P2A_03 and P3A_03 are more established than other routes, they have been selected to represent the BPI market.

This paper collects daily freight rates for six routes from January 4, 2016 to December 22, 2023. The Baltic Sea Exchange did not record transactions on non-working days, resulting in a total of 1,995 days and generating a dataset of 1, 995*6 = 11, 970 observations. The data were sourced from the Clarkson website (https://sin.clarksons.net/). The analysis was conducted using Eviews, R, and MATLAB. The current routes for Capesize and Panamax dry bulk carriers are presented in Table 2, while the time series plots of freight rates for the main routes in the BCI and BPI markets are illustrated in Figs 3 and 4.

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Fig 3. Trend plots of freight rates for C2, C3 and C5 routes.

https://doi.org/10.1371/journal.pone.0315167.g003

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Fig 4. Trend plots of freight rates for P1, P2 and P3 routes.

https://doi.org/10.1371/journal.pone.0315167.g004

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Table 2. Capesize dry bulk carrier routes.

https://doi.org/10.1371/journal.pone.0315167.t002

Data preprocessing

As shown in Figs 3 and 4, among the Capesize routes, the freight rate for the C3 route consistently exceeds that of the other two routes at various times, exhibiting similar fluctuation trends, particularly with significant volatility during the period from 2020 to 2022. For the Panamax routes, the freight rate for the P2 route is slightly higher than those of the other two routes, also demonstrating notable fluctuations during 2020 to 2022. Overall, the freight returns for all six routes display considerable instability. To assess the stationarity of the series, we calculate the returns using the logarithmic first-order difference as follows: (13) where Y_t represents the logarithmic return of route, and P_t represents the freight rate at time t. In order to increase the stability of the data, the unified magnification is 100 times. Denote C2*, C3*, C5*, P1*, P2*, P3* represent the logarithmic return of freight rate for C2, C3, C5, P1, P2 and P3 routes, respectively.

Fig 5 displays the time series of logarithmic returns obtained by calculating the first-order difference of the freight rates. It is evident that the logarithmic returns for each route fluctuate around zero, showing similar patterns of volatility clustering coinciding with significant events such as the US-China trade disputes, the COVID-19 pandemic, and the Russia-Ukraine conflict. However, the responses of different routes to these extreme shocks have varied over time. These characteristics present an opportunity to explore risk spillovers between the freight rates in the BCI and BPI markets.

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Fig 5. Trend plots of freight rate returns for each route in BCI and BPI markets.

https://doi.org/10.1371/journal.pone.0315167.g005

Nonlinear analysis of returns series for each route

Fig 6 presents scatter plots and histograms of the logarithmic returns for the various routes. The figure reveals a significant nonlinear correlation between the logarithmic returns of each route. As a result, traditional linear methods may not adequately capture the relationships between the routes. It is essential to employ a method that is suitable for nonlinear relationships in modeling. The Copula function effectively addresses linear constraints, facilitating research into nonlinear correlations.

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Fig 6. Scatter plots and sequence histogram of freight rate returns for each route.

https://doi.org/10.1371/journal.pone.0315167.g006

Descriptive statistics of returns series for each route

In Table 3, the mean logarithmic return for each route ranges from approximately 0.06 to 0.08, with a maximum value of 41.428 for C2* and a minimum value of -24.016 for C5*. The standard deviation for P1* is the highest, indicating significant fluctuations, while P2* has the lowest standard deviation, suggesting relatively stable fluctuations. The skewness values for all logarithmic returns are greater than zero, indicating that the distributions are not symmetrical and are skewed to the right. Additionally, the kurtosis values for all logarithmic returns exceed 3, signifying that these returns have higher peaks and thicker tails. The Jarque-Bera (J-B) test statistics are relatively large, and the corresponding p-values are less than 0.01, indicating that the logarithmic returns do not follow a normal distribution. The Augmented Dickey-Fuller (ADF) test results suggest that the sequences are stationary. Furthermore, the Q(12) test indicates no autocorrelation among the various return sequences at a significance level of 1% with a lag of 12 orders.

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Table 3. Descriptive statistics of freight rate returns for each route (p-value).

https://doi.org/10.1371/journal.pone.0315167.t003

Endogeneity and heteroskedasticity tests

The Granger causality test helps identify the causal relationships between variables and assess endogeneity. As shown in Table 4, there is a one-way Granger causality from C2 to P2, C2 to P3, C3 to P2, C3 to P3, and C5 to P3 at a significance level of 10%. The remaining relationships exhibit two-way Granger causality. This indicates that the lagged values not only influence their own future outcomes but also provide statistically significant information to predict the outcomes of other markets, demonstrating a causal effect on other market sequences. These findings offer a crucial basis for understanding the linkages between freight rates across various routes.

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Table 4. Results of Granger causality test.

https://doi.org/10.1371/journal.pone.0315167.t004

The ARCH effect test is a crucial step in evaluating the heteroscedasticity of residuals after modeling high-frequency time series data. In this paper, we employ the LM test statistic to assess the ARCH effect. The results of the ARCH effect test for the residual variances of each sequence are presented in Table 5. As indicated by the p-values in the table, we reject the null hypothesis at a significance level of 0.05, confirming the presence of an ARCH effect in the logarithmic returns of each series. Each logarithmic return series exhibits characteristics such as sharp peaks, thick tails, non-normality, and conditional heteroscedasticity.

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Table 5. Results of ARCH effect test.

https://doi.org/10.1371/journal.pone.0315167.t005

Edge distribution estimation

The primary purpose of using the GARCH model is to effectively capture the heteroscedasticity present in the logarithmic return series of freight rates across various routes, specifically the characteristics of volatility changes over time. When establishing a GARCH model, it is essential to make reasonable assumptions regarding the distribution of the residual perturbation terms in the series. Typically, residuals can be assumed to follow various distribution types, including the normal distribution, t-distribution, and skewed t-distribution. Therefore, we will examine the GARCH-normal model, GARCH-t model, and GARCH-skewed t model separately. We select the GARCH (p,q) order and residual distribution based on the AIC and Log Likelihood function (LLF), while passing the significance test of the parameters and considering the simplicity of the model.

Table 6 presents the GARCH(p,q) results for each sequence with various residual distributions, with the GARCH(1,1)-skew-t model ultimately selected to estimate the edge distribution. To mitigate the impact of a high-order mean model on subsequent modeling, ARMA(1,1) is consistently chosen under the constraint of sequence autocorrelation. Consequently, all sequences utilize the ARMA(1,1)-GARCH(1,1)-skew-t framework to characterize volatility.

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Table 6. GARCH modeling and evaluation of freight rate returns for each route.

https://doi.org/10.1371/journal.pone.0315167.t006

In Table 7, all values of α₁ + β₁ are less than 1, which ensures the effectiveness of the GARCH model in fitting the fluctuations of freight rates for the internal routes in BCI and BPI markets. Except for C2* series, all other β₁ is greater than the corresponding α₁, which indicates that they all have strong and sustained fluctuations during the sample period. The values of α₁ + β₁ for C3* series and P1* series are very close to 1, which indicates that these two returns series have higher sustained volatility and long memory than other series. The values of shape range from 2 to 5, and all p-values are very close to 0, indicating that the degree of freedom parameter is suitable for data with significant skewness and thick tails. The p-values of ARCH-LM test are all greater than 0.1, which indicates that the ARCH effect no longer exists in these series. That is, the GARCH(1,1)-Skew-t model eliminates the ARCH effect in each series (ARCH-LM test uses 10th order). Overall, the model and edge distribution selected here are appropriate.

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Table 7. ARMA(1,1)-GARCH(1,1)-skew-t modeling parameters (p-value).

https://doi.org/10.1371/journal.pone.0315167.t007

Nonlinear correlation analysis

Static Copula correlation analysis.

After estimating the GARCH edge distribution, we can obtain the conditional mean, conditional standard deviation, and standardized residuals of the sequence. The correlation between paired series will be derived from the probability integral transformation in the BCI and BPI markets using binary Copula modeling. The most suitable Copula function is selected based on the criterion of minimizing the Akaike Information Criterion (AIC).

In Table 8, all pairs, except for C2*-P2*, C3*-P2*, and C5*-P1*, utilize the t Copula as the optimal function, indicating a degree of homogeneity among these routes. The correlations for C2*-P1* and C3*-P1* are relatively low, which may be attributed to the impact of route locations on their correlation. The P1 route connects Europe and America along the west coast of the Atlantic, while the C2 route links Brazil with Western Europe, and the C3 route connects Brazil to China. With the exceptions of C3*-P2* and C5*-P1*, there is a symmetric positive correlation in the tails of all other routes. Overall, a positive correlation exists among the various routes.

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Table 8. Copula functions modeling parameters.

https://doi.org/10.1371/journal.pone.0315167.t008

The largest static correlation is observed in the C5*-P3* pair, while the relatively small static correlations are found in C5*-P1*, C3*-P1*, and C2*-P1*. The linkage between the freight rate of the BCI market and the P1 route is notably low, which may be attributed to the varying impacts of different voyages on the correlation. The P1 route primarily serves as an Atlantic round-trip route, originating in the United States and terminating in Western Europe. In contrast, the C2, C3, and C5 routes do not originate from the United States, with both the C3 and C5 routes ultimately heading to China.

Dynamic Copula correlation analysis.

To effectively illustrate the variation of the correlation coefficient over time, three types of time-varying Copula functions are selected to analyze the correlation between different freight rates. The parameters of these time-varying Copula functions are dynamically adjusted. This study employs the time-varying Normal (TVN) Copula, time-varying Rotated Gumbel (TVRG) Copula, and time-varying SJC (TVSJC) Copula for dynamic Copula modeling.

In Table 9, the time-varying SJC Copula is the optimal choice for all pairs except for C2*-P3*. The parameters of the time-varying SJC Copula are especially sensitive to tail data, allowing it to effectively capture asymmetric correlations in both the upper and lower tails. However, since the time-varying SJC Copula primarily focuses on characterizing tail correlation coefficients, we also employ the time-varying Normal Copula to better represent the overall symmetric correlation between route fares.

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Table 9. AIC values of different dynamic Copula functions.

https://doi.org/10.1371/journal.pone.0315167.t009

From Table 10, it is evident that, except for C2*-P2*, C2*-P3*, and C5*-P2*, all other pairs exhibit positive correlations. Notably, C2*-P3* shows a few negative correlations during the early stages, which may be linked to the US interest rate hike in December 2015 and the collapse of the Chinese stock market around New Year’s Day in 2016. Both C2*-P2* and C5*-P2* display individual negative correlations in the later period, potentially due to the impacts of COVID-19. Examining the fluctuations in dynamic correlations, the standard deviations for C2*-P2* and C2*-P3* are relatively large, indicating frequent fluctuations. In contrast, the standard deviation of C5*-P1* is only 0.015, suggesting a relatively stable variation throughout the sample period.

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Table 10. Descriptive statistics of time-varying Copula dynamic correlation coefficients.

https://doi.org/10.1371/journal.pone.0315167.t010

The time series of dynamic correlation coefficients is illustrated in Fig 7. C5*-P3* demonstrates the largest mean dynamic correlation, while C5*-P1* shows the smallest mean dynamic correlation. This finding aligns with the conclusions drawn from the static optimal Copula analysis, indicating that the time-varying Normal Copula modeling performs well in capturing these relationships.

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Fig 7. Trend plots of the TVN Copula dynamic correlation coefficients between the BCI and BPI markets.

https://doi.org/10.1371/journal.pone.0315167.g007

To further analyze the dynamic correlation between the upper and lower tails of freight rates for various routes in the BCI and BPI markets, we will employ the TVSJC Copula. As illustrated in Figs 8–10, the dynamic correlation coefficients exhibit asymmetric dependency structures. For all six pairs, the mean values of the lower tail correlation coefficients exceed those of the upper tail correlation coefficients. This suggests that the correlations between freight rates are stronger during market downturns or extreme crises. Additionally, there is a higher likelihood of simultaneous price drops while experiencing different price increases across freight routes in the BCI and BPI markets.

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Fig 8. Trend plots of TVSJC Copula dynamic correlation coefficients between the C2* and BPI markets.

https://doi.org/10.1371/journal.pone.0315167.g008

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Fig 9. Trend plots of TVSJC Copula dynamic correlation coefficients between the C3* and BPI markets.

https://doi.org/10.1371/journal.pone.0315167.g009

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Fig 10. Trend plots of TVSJC Copula dynamic correlation coefficients between the C5* and BPI markets.

https://doi.org/10.1371/journal.pone.0315167.g010

Empirical analysis of risk spillover effect

Dynamic time-varying VaR results.

Before calculating CoVaR, it is essential to first compute VaR, as defined by the formula in (4). Fig 11 illustrates the dynamic time series of returns for each series. Table 11 presents the descriptive statistics of the time-varying VaR for each series.

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Fig 11. Trend plots of VaR for each returns pairs.

https://doi.org/10.1371/journal.pone.0315167.g011

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Table 11. Descriptive statistics of time-varying VaR for each series (α = 0.05).

https://doi.org/10.1371/journal.pone.0315167.t011

From Fig 11 and Table 11, it is evident that the VaR of each route’s freight rates exhibits time-varying characteristics with notable fluctuations. In the BCI market, the C3 route has a lower risk value compared to the other two routes, while the absolute VaR for the C5 route is the highest. This difference may be attributed to the C5 route, which runs from Western Australia to Qingdao, whereas the C3 route connects Tubarang (Brazil) to Qingdao. Australia consistently ranks first in global iron ore export volume, significantly surpassing Brazil and other regions. Consequently, trade fluctuations within the Australian domestic market and variations in iron ore exports may introduce potential risks to the freight rates of the C5 route.

In the BPI market, the P1 route exhibits the largest absolute value of average VaR, indicating that its fluctuations are the most significant. Conversely, the P2 route has the smallest absolute value of average VaR, suggesting its fluctuations are the least pronounced. This disparity may be attributed to the P2 route, which connects the Atlantic with the Far East (including China, Japan, and East Asia). In recent years, both China and Japan have implemented shipping trade policies that partially mitigate freight rate risks. Notably, the risk fluctuations for each route significantly increased in 2020 and 2022, likely due to the impacts of the global COVID-19 pandemic in 2020 and the Russia-Ukraine conflict in 2022. This underscores the influence of major global events on freight rates.

Dynamic time-varying CoVaR results.

CoVaR measures the magnitude of mutual risk spillover effects between the BCI and BPI markets. Table 12 presents the descriptive statistics for the dynamic time-varying CoVaR of the BCI market as it relates to the BPI market, while Table 13 provides similar statistics for the BPI market in relation to the BCI market. From Tables 12 and 13, it is evident that the risks associated with the BCI and BPI routes have experienced different changes over time. Notably, the absolute value of the average VaR for the C2 route is 4.2383, as shown in Table 11, while the absolute value of the average CoVaR^P1|C2 is 5.1869, the absolute value of the average CoVaR^P2|C2 is 6.4045, and the absolute value of the average CoVaR^P3|C2 is 5.2595, both of which exceed the VaR of the C2 route. This suggests that the VaR for the C2 route is likely underestimated, highlighting a limitation of relying solely on VaR to assess the risk of return series. Consequently, CoVaR is employed to measure the tail losses associated with VaR, providing a more comprehensive risk assessment.

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Table 12. Descriptive statistics of dynamic time-varying CoVaR in BCI → BPI (α = 0.05).

https://doi.org/10.1371/journal.pone.0315167.t012

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Table 13. Descriptive statistics of dynamic time-varying CoVaR in BPI → BCI (α = 0.05).

https://doi.org/10.1371/journal.pone.0315167.t013

To further illustrate the difference between VaR and CoVaR, we present the upward and downward risks of two markets. Fig 12 depicts the risk spillover effect of the BCI market on the BPI market, while Fig 13 shows the spillover effect of the BPI market on the BCI market. From Figs 12 and 13, it is clear that the VaR values for each route, as well as the CoVaR values for each pair, exhibit dynamic changes over time. Notably, the CoVaR from C5* to P1* largely overlaps with the VaR of P1*, indicating a one-way risk spillover from the freight rate of the P1 route to the freight rate of the C5 route. Conversely, the freight rate of the C5 route does not spill over risk to the freight index of the P1 route. Similarly, there is no spillover effect between the freight rates of the C3 and P2 routes. Additionally, the absolute values of the upward and downward CoVaRs are greater than their corresponding VaRs, which highlights the presence of ΔCoVaR and confirms the existence of bidirectional risk spillover.

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Fig 12. Time series plots of dynamic CoVaR and VaR (BPI→ BCI).

https://doi.org/10.1371/journal.pone.0315167.g012

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Fig 13. Time series plots of dynamic CoVaR and VaR (BCI → BPI).

https://doi.org/10.1371/journal.pone.0315167.g013

Dynamic time-varying ΔCoVaR results.

This paper focuses on studying risk spillovers in extreme loss situations, specifically exploring the dynamic time-varying ΔCoVaR relationship between two markets under severe downward scenarios. The calculation formula for ΔCoVaR is provided in (6). Table 14 presents the descriptive statistical results for the dynamic time-varying ΔCoVaR from the BCI market to the BPI market, while Table 15 details the results from the BPI market to the BCI market. To better illustrate the intensity of risk spillover between different routes in the two markets, we include a time-varying plot of bidirectional risk spillover ΔCoVaR, as shown in Fig 14. From Tables 14 and 15, along with Fig 14, we can observe the following:

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Fig 14. Time series plots of ΔCoVaR (α = 0.05).

https://doi.org/10.1371/journal.pone.0315167.g014

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Table 14. Descriptive statistics of dynamic time-varying ΔCoVaR for freight rate returns (BCI→BPI).

https://doi.org/10.1371/journal.pone.0315167.t014

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Table 15. Descriptive statistics of dynamic time-varying ΔCoVaR for freight rate returns (BPI→BCI).

https://doi.org/10.1371/journal.pone.0315167.t015

Firstly, the ΔCoVaR of freight rates for each route exhibits temporal fluctuations and some extreme values, highlighting the limitations of VaR in capturing risk spillovers. This confirms that VaR does not fully measure the complexities of risk interconnections.

Secondly, the risk spillover from P2 to C2 is significantly greater than that from C2 to P2. Similarly, the risk spillover from P2 to C5 exceeds that from C5 to P2, and the risk spillover from P3 to C5 is greater than that from C5 to P3. Notably, C5 has a relatively small risk spillover effect on P1, indicating a one-directional spillover between P1 and C5. This finding aligns with conclusions drawn in the previous section. With the exception of C5 and P1, all other pairs demonstrate two-directional and asymmetric risk spillover effects.

Thirdly, the absolute mean risk spillover from C3 to P1 is the largest. This can be attributed to the geographical proximity of the routes; both C3 and P1 are linked to the Atlantic, while C5 traverses the Pacific. Conversely, the mean risk spillover from C5 to P1 is the smallest, likely due to the greater distance between their respective starting points. Specifically, the distance of the C3 route is more than twice that of the C5 route. The mean risk spillover from P2 to C2 is the largest, while that from P1 to C5 is the smallest, further suggesting that route distances play a significant role in determining the magnitude of risk spillovers.

Fourthly, the ΔCoVaR values between -12 and 0 indicate that downward risk spillovers are consistently negative, and there is a positive correlation in the direction of risk between each freight rate pair. This suggests that freight rates in both the BCI and the BPI tend to move together, whether rising or falling.

Fifthly, although the time-varying ΔCoVaR plots differ among the routes, there is a clear trend: the absolute values of ΔCoVaR significantly increased from 2020 to 2022. This increase can be attributed to external shocks such as the COVID-19 pandemic and the Russia-Ukraine conflict, which have notably impacted risk spillovers between routes. This observation supports the argument that extreme events can substantially alter spillover effects.

Dynamic time-varying %CoVaR results.

Although ΔCoVaR effectively measures risk spillover effects, it does not eliminate the influence of dimensionality. Therefore, we will proceed to calculate %CoVaR. This metric represents the relative strength of risk spillover effects and reflects the contribution of these effects between freight indices in two markets under extreme risk conditions. Table 16 presents the descriptive statistics for the downward dynamic time-varying %CoVaR from the BCI market to the BPI market, while Table 17 provides similar statistics for the downward dynamic time-varying %CoVaR from the BPI market to the BCI market. Additionally, to offer a clearer understanding of the intensity of risk spillover between each return pair, the rankings of the mean values for %CoVaR are displayed in Table 18.

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Table 16. Descriptive statistics of dynamic time-varying %CoVaR for freight rate returns (BCI→BPI).

https://doi.org/10.1371/journal.pone.0315167.t016

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Table 17. Descriptive statistics of dynamic time-varying %CoVaR for freight rate returns (BPI → BCI).

https://doi.org/10.1371/journal.pone.0315167.t017

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Table 18. %CoVaR mean ranking result (α = 0.05).

https://doi.org/10.1371/journal.pone.0315167.t018

From Tables 16–18, we can conclude that:

Firstly, all %CoVaR values are positive, indicating that extreme conditions affect both markets. Furthermore, there are significant differences in the intensity of risk spillovers among various return pairs.

Secondly, regarding the risk spillover from the BCI market to the BPI market, the freight rate of the C5 route contributes the most to the risk of the P3 route. This is likely due to the fact that both routes are connected to East Asia. Conversely, the C5 route has the least contribution to the risk spillover of the P1 route, primarily because there is less overlap between these two routes, which aligns with the findings of ΔCoVaR.

Thirdly, in the risk spillover from the BPI market to the BCI market, the freight rate of the P2 route has the greatest impact on the risk intensity of the C2 route. This relationship may be attributed to the distance of the routes; the P2 route is the longest among the three BPI internal routes, and longer distances tend to increase risk spillover intensity. On the other hand, the risk intensity of freight rates between the P1 route and the C5 route is the lowest, likely due to a lack of intersection between the two routes.

Finally, although the risk intensity from the P1 route to the C5 route is greater than that from the C5 route to the P1 route, the overall risk spillover intensity between these two routes is significantly lower than that of all other pairs.