An innovative tool for cost control under fragmented scenarios: The container freight index microinsurance

Abstract

With the impact of the COVID-19 pandemic, global container freights have increased dramatically since the second half of 2020, which has significantly hampered the booking activities of fragmented transportation space for small and medium-sized import and export enterprises (SMIEEs). To provide SMIEEs with an effective tool for controlling shipping costs, we propose the design principles of index microinsurance under fragmented scenarios and design the container freight index microinsurance (CFIM) based on a comprehensive analysis of the term, compensation and share structures. We further establish the pricing model for the CFIM and selection procedure for product optimization, and illustrate the framework with a case study based on the data of the China Containerized Freight Index Europe Service, which demonstrates the good performance of the designed product even under extreme market conditions. The design principles proposed can shed light on the innovation of index microinsurance product that meets fragmented needs and the newly designed CFIM, along with the pricing and optimization procedure, provides practitioners with useful tools for cost control.

1. Introduction

Container maritime transportation plays an irreplaceable role in the worldwide trading system (Stopford, 2008). In recent years, with the rise of global cross-border e-commerce, a large number of small and medium-sized import and export enterprises (SMIEEs) have joined the global maritime supply chain system, and a new trend of international logistics characterized by high-frequency dispersed replenishment and short-term spot freight contract has been formed. According to Southern Africa’s freight news, the fragmented less container load (LCL) business accounted for nearly 20% of the total global container throughput in 2021. The increase in economic uncertainties and decrease in global trade orders have brought great challenges to SMIEEs. Large fluctuations in container freight can cause increases in logistics costs, and hence reduce meager profits for SMIEEs.

Due to the impact of the recent COVID-19 pandemic, the global container liner shipping market has been experiencing steep surges in freight costs since the second half of 2020, and the freights of some liner routes such as Asia-US East Coast have even increased by 400%. The China Containerized Freight Index (CCFI) reached a historic peak of 3047.32 on Aug. 20, 2021, with an increase of 215% from last year. The increase in logistic costs has undoubtedly pushed SMIEEs further toward plight. Consequently, a large number of SMIEEs, which are in a weak position in the procurement of short-term spot freight contracts, urgently need a tool for freight risk management and cost control.

However, there are still no effective market-oriented tools for managing the risk of container freight fluctuations faced by SMIEEs. On the one hand, traditional trading tools including Free on Board (FOB), Cost and Freight (CFR), Cost, Insurance and Freight (CIF), and Long Term Contract (LTC), are only limited to the allocation between two trading parties, and cannot effectively reduce the cost through risk transitions. On the other hand, there is no effective financial derivative for container freights, which makes SMIEEs helpless towards fluctuations of container freights. Previous studies indicate that financial derivatives can be used to significantly reduce business costs and financial risks for non-financial enterprises (Gay et al., 2011, Ahmed et al., 2018) and reduce logistic costs and increase firm value in the shipping industry (Kavussanos and Visvikis, 2006, Tsai et al., 2009). From a practical perspective, the development of global container freight derivatives is far behind the development of dry bulk cargo and tanker freight derivatives, which is unfavorable to controlling the additional costs caused by the fluctuation of freights. The Shanghai Shipping Exchange launched container freight index futures in 2011, but the daily trading volume is still quite small, only several hundred contracts, because of the regulatory constraints of the China Securities Regulatory Commission. Actually, it is difficult for SMIEEs to participate in the trading of container freight derivatives because of the requirement of large face value, trading capital and advanced skills.

In this paper, we propose a new financial tool, i.e., the container freight index microinsurance (CFIM) to help SMIEEs manage the freight rate risk and control costs. To ensure the sustainability of the product business model, our design of the CFIM is both market-oriented and inclusive, which means that the CFIM meets both the demand of SMIEEs and the operationality of insurers. Index microinsurance is a new application of the existing index insurance under fragmented scenarios, which requires new design principles to reflect the new features. The existing index insurance products are mainly designed for the benefit of insurers, thus leading to excessively high rates and harsh underwriting conditions. In addition, most of the existing index insurance products overlook the demands of fragmented insurance scenarios such as small-amount and high-frequency coverage. To address this problem, in the context of the development of small-amount and high-frequency fragmented insurance scenarios, we propose the design principles of index microinsurance for the fragmented scenario from the interest of both insurers and policyholders. Based on the principles, we design the CFIM to provide a financial tool for SMIEEs to manage the risk of container freight cost fluctuations at reasonable prices.

The main contributions of this paper are as follows: To the best of our knowledge, we are the first to propose the design principles of index microinsurance in fragmented insurance scenarios, and design the CFIM for SMIEEs based on the proposed principles. The CFIM designed has unique term, compensation and share structures, meeting the risk management needs of both SMIEEs and insurers. We also derive the pricing model and establish the selection procedure for optimal product designs. The whole framework proposed not only fills the gaps in the existing literature but also provides practitioners with useful tools for risk management.

The rest of the paper is structured as follows: Section 2 reviews the relevant literature. Section 3 describes the design principle of CFIM. Section 4 presents the details of the design procedure including pricing. Section 5 illustrates the proposed framework with a case study, and Section 6 concludes the paper.

2. Literature review

Due to the distinctive characteristics of extremely dramatic fluctuations in maritime freights, how to reduce the risk of freight fluctuations by means of freight futures, freight options and other shipping derivatives has drawn widespread attention. The container freight index insurance is a contingent right in the future, essentially the same as an option. Therefore, the hedging strategy and pricing model of freight options can serve as the theoretical base of our study.

One branch of the literature mainly focuses on the strategies and efficiency of hedging freight fluctuations with the forward freight agreement (FFA) and freight futures in the dry bulk cargo and tanker marine sub-market. Many of these studies examine the optimization of the hedging strategy of freights. For example, Tezuka et al. (2012) derived the equilibrium spot price and forward/futures curve formulae for freight shipping markets by generalizing the Bessembinder and Lemmon (2002) model and discussed the properties of the forward risk premium and the optimal hedge ratio that corresponds to the reduced spot freight risk. Prokopczuk (2011) empirically compared the pricing and hedging accuracy of a variety of continuous-time futures pricing models with single-route dry bulk freight futures contracts and showed that the inclusion of a second stochastic factor significantly improved the pricing and hedging accuracy. Nomikos and Doctor (2013) used the Sharpe ratio to study the dynamic hedging strategies for different FFA contracts and different maturities. Hsu et al. (2015) developed tanker freights FFA hedging strategies with the bivariate asymmetric non-linear smooth-transition GARCH method. Adland and Alizadeh (2018) claimed that when hedging a vessel's earnings, the hedging ratio of FFAs could be adjusted according to the related factors of the vessel and the contract. Gu et al. (2020) evaluated the quantile hedge ratios of the FFAs, and found that the hedging performance tends to be different from the approach of minimum variance. Sun et al. (2018) proposed an optimal combination of hedging strategies for trading the derivatives of crude oil futures and dry bulk FFA simultaneously with the cross-market dynamic relationship. There are also studies about the hedging efficiency of freight futures. For example, Goulas and Skiadopoulos (2012) investigated whether the dry and wet freight futures market was efficient over the daily and weekly horizons, and found that those were not efficient over the shorter daily horizon. Adland et al. (2020) investigated the corresponding hedging efficiency when using a portfolio of FFA prices to hedge the ship price risk of both static and dynamic hedge ratios, indicating that the hedging efficiency was greater for newer vessels than older vessels and that the static ratio outperformed the dynamic ratio. Bai et al. (2022) investigated the effectiveness of financial hedging strategies of tramp shipping companies and concluded that financial hedging can effectively reduce bunker fuel prices but not freights. In contrast, few studies have explored the hedging strategy or efficiency of container freight futures.

Another branch of the literature mainly concentrates on improving the accuracy of freight options pricing for hedging the freight fluctuation with freight options. Since most freight options belong to path-dependent Asian options, many scholars have studied pricing models and accuracy. In the early days, the pricing model of freight option is constructed assuming that the price of the underlying freight follows a lognormal distribution. For instance, Koekebakker et al. (2007) constructed a freight fitting model based on lognormal distribution, and deduced the closed pricing formula of Asian call and put options with fixed execution prices, which significantly improved the pricing and hedging accuracy. Based on Monte Carlo simulation, Wang et al. (2009) constructed the Asian option pricing model with the lognormal distribution and fixed strike price. Haug (2021) gave the Turnbull and Wakeman (1991) adjustment closed formula to calculate the settlement price of freight options on the European Energy Exchange. Previous studies found that the invalidity of the lognormal distribution assumption will distort the freight option price. Therefore, it is necessary to further improve the underlying freight distribution assumption in order to get a more accurate pricing model. Nomikos et al. (2013) built a valuation approach for options on the average spot freight by extending the traditional lognormal representation for the risk-neutral spot freight dynamics to a diffusion model overlaid with jumps. Kyriakou et al. (2017) used the exponential mean-reverting diffusion model with a decay factor to price the fixed strike price of the freight option, and calculated the price of the freight option and the dynamic changes under different diffusion levels, which demonstrates improvement in reducing pricing bias. Gómez-Valle et al. (2020) proposed a freight option pricing model based on stochastic delay partial differential equations and gave the cap and floor limits of the option price. Besides, there are also studies discussing the characteristics of freight options. Alexandridis et al. (2017) analyzed the relationship of economic spillovers among spots, futures, and options. Lim et al. (2019) analyzed the drivers of freight market volatility and examined their impact factors on the term structure of freight options implied volatilities. It can be seen that due to the complexity of freight, the pricing accuracy of freight options largely depends on the validity of the assumptions. In addition, the pricing of more complex freight option portfolios remains unexplored.

The customers of the dry bulk and tanker market are mainly large import and export enterprises, while SMIEEs are taking up more shares in the container market. The existing literature about maritime freight derivatives has limitations as follows. The small-amount and high-frequency characteristics of SMIEEs’ demands for freight risk management require specialized product design with unconventional term, compensation and share structures, which has not been addressed by previous studies. Also, the complicated product design requires more advanced pricing modeling and optimization procedure, which has not been fully covered in the literature.

3. Design principles of container freight index microinsurance

This is a true story experienced by many SMIEEs. On Oct.16, 2020, YR, a small toy manufacturer in Wenzhou, China, obtained an order for children's toys from a European customer due in 3 months. The contract value is $10,000, and YR beard the cost of linear freight. YR's estimated net profit rate was 3%. As production and transportation each take about 1 month, YR had about 1 month for booking the liner shipping space. On Oct.17, YR inquired about two pieces of information: the container freight from China to Europe was $1082 /TEU, and the pallets needed was 0.5 TEU with a cost of $550. The shipping cost accounted for 5.5% of the contract transaction amount. When YR completed production and started ordering container space, the container shipping market underwent a dramatic change. The freight quoted the next day reached $1324 /TEU, and the pallet container freight increased to $700 with a rising trend. Due to the significant increase in container freight, the actual profit rate became 3%−(700/550−1)*5.5% = 1.5%. After investigation, we find that none of the container transportation, futures and insurance markets have provided effective risk management tools to help SMIFEs cope with the increased financial pressure caused by rising freight costs. The impact of COVID-19 further increases the need of such tools.

Indeed, with the in-depth development of digitalization and technology in various industries in recent years, fragmented insurance scenarios featured with small-amount and high-frequency transactions have an increasingly frequent presence, which is particularly the case in China. Such fragmented insurance scenarios emphasize affordable or inclusive insurance and the interests of insurance policyholders. In contrast, traditional index insurance is basically designed for risk management from the perspective of insurers (Goodrich et al., 2020, Assa and Wang, 2021), which may have adverse effects on potential policyholders, such as high premiums, low leverage and harsh underwriting conditions, and hence filter out the consumers who have actual needs.

Our proposed index microinsurance highlights market orientation and intrinsic inclusiveness. This requires that the design principles should reflect the need of both the supply side and the demand side. Consequently, we propose the four principles of index microinsurance: reliable index, adaptive structure, affordable price, and controllable risk, hereinafter referred to as the RAAC principles.

As the CFIM is designed for risk management in fragmented insurance scenarios, our proposed principles should be followed as elaborated below.

Principle 1, reliable index. The underlying index of an index insurance or index derivative should be transparent and objective. If the index information is not transparent and publicly accessible, it can be easily manipulated. When index insurance is based on an opaque index, the moral hazard will be very large (Goodrich et al., 2020). To ensure higher risk transfer efficiency and lower replication cost, the compilation of the index should aim at capturing the true trajectory of the target and reducing the underlying basis risk (Li et al., 2021). The wide application of weather index insurance highlights the need for transparency and objectivity for the underlying index (Hohl et al., 2020). Therefore, the CFIM should be built on a reliable container freight index.

Principle 2, adaptive structure. In practice, the index insurance should be designed to match the needs of potential customers for risk management including the period and exposure to be covered. Also, the term, compensation and share structures should be adaptive to the actual needs of consumers. Products tailored to customers' needs are more easily accepted by the market (Jensen et al., 2019). As for the CFIM, the structure should meet the cost transfer requirements of SMIEEs for container booking activities. More specifically, the term structure should match the most important and common term structure in the booking activities of SMIEEs. Adaptive compensation mechanisms such as hierarchical compensation and limit compensation should be designed according to whether the index is specifically triggered or not at specific frequency and severity levels. Considering the fragmentation characteristics of SMIEEs to book container space, the basic share or minimum share in a container should be set.

Principle 3, affordable price. Most index insurances belong to inclusive insurance (Institute of International Finance, 2018). Different from exchange-traded financial derivatives, there are no entrance barriers for purchasing index insurances, which makes them accessible to all kinds of consumers including SMIEEs (Assa and Wang, 2021). Fragmented risk management requires the price should be affordable, enabling potential customers to obtain risk protection or cost transfer without significantly increasing their economic burden. Therefore, it is important that the CFIM is designed in line with the current development trend of inclusiveness and minimalist of financial derivatives, and is featured with lower insurance premium rates and higher financial leverage to benefit many SMIEEs.

Principle 4, controllable risk. Index insurance is essentially a claim game around the specific frequency and severity of the fluctuation of the underlying index, which requires the insurance product to be designed under sufficient considerations of risk controls and sustainable operation (Taib and Benth, 2012, Conradt et al., 2015, Arandara et al., 2019, Leblois et al., 2020, Ceballos and Robles, 2020). The design of the CFIM should also follow this principle. For example, the key parameters used in insurance pricing, such as the volatility and triggering target price, should be set prudently. When adjusting the underwriting scale should be adjusted dynamically according to the market volatility. The safety buffer of the insurer should be sufficient to deal with extreme fluctuations in the container freight index. In brief, a well-designed insurance product can not only help SMIEEs transfer risks under different conditions but also help insurers achieve reasonable profits and commercial sustainability.

The RAAC principles of the index microinsurance design are mutually complementary and indispensable: reliable index guarantees the objectiveness of the product; adaptive structure reflects the specialized design; affordable price affirms the product inclusiveness, and controllable risk meets the risk control requirements of the insurer.

4. Design of container freight index microinsurance

In this section, we will introduce the basic framework of CFIM design and discuss about the key procedures including index selection, microinsurance structure, pricing model, and scheme selection criteria.

4.1. The basic framework

In this paper, we propose a design of the CFIM based on the RAAC principles, taking into account the needs and interests of both the insurer and SMIEEs. The design consists of the following steps. First, we select the container freight index to ensure the reliability of the developed products. Second, we design the compensation structure, term structure (including without compensation cap and with compensation cap) and basic share structure of CFIM, and the scheme set of initial microinsurance products under the combination of various parameters such as the target point of compensation and insurance amount. Third, inspired by the forward-starting arithmetic average Asian call option pricing model of Turnbull and Wakeman (1991), we set up two microinsurance pricing models of container freight index compensation without and with a cap to determine the premium rate of product schemes. Last, we construct criteria such as the average net premium rate (ANPR) per basic share and cumulative underwriting profit ratio (CUPR) to select the optimal scheme of CFIM products.

4.2. Index selection

According to the principle of reliable index, the container freight index with strong reliability should be adopted. The China Containerized Freight Index (CCFI) and Shanghai Containerized Freight Index (SCFI) compiled by the Shanghai Shipping Exchange, the Freightos Baltic Container Index (FBX) compiled by the Baltic Exchange, the World Container Freight Index (WCI) compiled by the Drewry, and the Platts Container Freight Index system of Reuters are the prevailing indices in the global maritime container transportation market. Reflecting the huge volume of container freight in the world's leading trading economy, the CCFI has become an important reference for daily operations for increasingly more participants in the global shipping market (Hsiao et al., 2014, Kim et al., 2016). The CCFI is a weekly index mainly compiled from the average price information of spot market booking for general shippers provided by 22 shipping companies with outstanding reputations and large market shares. The index covers 10 ports of departure in China including Dalian, Tianjin, Qingdao, and Shanghai, and 12 line services from Europe, the United States and Japan. The index-linked agreement and index derivatives trading based on the CCFI have led to innovative pricing and trading mode in the shipping industry. It is clear that the CCFI meets the principle of reliable index, and hence is selected as the index for the CFIM.

4.3. Microinsurance structure

The story of YR introduced above vividly illustrates the traditional and typical trading process of SMIFEs. As the forward-starting booking is conventionally adopted, SMIEEs follow the three-stage procedure: (1) Space inquiry. After taking trading orders, SMIEEs make inquiries to third-party shipping agents and container shipping companies for the liner shipping space of the transport service. (2) Booking and payment. Booking orders are used to lock the space in the container before the departure time of the order goods. Noteworthily, most of the trade orders of SMIEEs are less container load (LCL) rather than full container load (FCL). For example, the pallet of YR is only 0.5 TEU, so it belongs to LCL. (3) Freight departure. The shipment will be launched as scheduled. During the process, SMIEEs will bear the burden of additional contingent costs caused by the increase in container freights. The newly proposed CFIM is designed to solve this problem. It works as follows: if the settlement point of a container freight index in a time window before expiration is higher than the target point of the freight index agreed between the insurer and SMIEEs, the insurer will cover the related costs; otherwise, the insurer pays nothing. The term, compensation and share structures of the CFIM are elaborated below.

Following the principle of adaptive structure, we further specify the structures of the insurance term, compensation and basic share as below.

4.3.1. Term structure

The term structure of microinsurance is designed as the forward-starting mode following a conventional practice. The term of the insurance contract, denoted as $T$, is set to cover the period between space inquiry and freight departure, and the insurance contract officially takes into effect with the beginning of the shipping space inquiry. The insurance term can be divided into the lock-up and valuation periods: The lock-up period covers the time period before booking, during which SMIEEs mainly consult about booking shipping space and follow up on the shipping price and space of container liners without actually making the booking. The valuation period is the key period during which SMIEEs will issue booking orders and determine the actual shipping cost (see Fig. 1 ).

Fig. 1.

The CFIM term structure.

4.3.2. Compensation structure

For the CFIM, we consider two types of compensation structures, i.e., with and without a compensation cap.

CFIM without compensation cap. At the expiration of the microinsurance, if the settlement point of the container freight index, denoted as $J$, is larger than the trigger target index point agreed in advance, denoted as $X$, the insurer will pay the amount $\theta (J-X)$ to the SMIEEs, where $\theta$ denotes the currency conversion coefficient of the freight index. Assuming that it is equally likely for SMIEEs to determine the freights at any time in the valuation period, the arithmetic average of the freight index in the valuation period is taken as the settlement point. If the settlement point is no larger than the target point, the insurer pays nothing. The risk exposure under CFIM without a cap is boundless for the insurer.

CFIM with compensation cap. The compensation structure is the same as that of the CFIM without a compensation cap except that, when the settlement point of the container freight index is larger than the trigger target index point, the insurer shall compensate the SMIEEs for the insured amount $G$ negotiated in advance by both parties. Under this compensation structure, the insurer is only subject to limited risk exposure.

As shown in Fig. 2 , under the compensation structure without a cap, SMIEEs can effectively transfer all additional logistics costs caused by the rise of freights to insurers, while the risk exposure of insurers is infinite. Under the compensation structure with a cap, the risk exposure of the insurer is bounded, though SMIEEs can only partially transfer the additional logistics costs.

Fig. 2.

Both types of compensation structures when the CFIM contract expires and settles.

4.3.3. Share structure

Standard container booking is mainly FCL or LCL. With the rapid development of cross-border e-commerce, it is more conventional for SMIEEs to adopt LCL. To meet the need of controlling transportation costs of small pallets of SMIEEs, we design the CMIF to make better use of the fragmented transport space. In practice, freight ton, the lowest charge standard, is often used for LCL. Freight ton consists of weight ton and volume ton: Weight ton is based on the gross weight of goods, with 1 metric ton as 1 freight ton, and volume ton is based on the gross volume of goods, with 1 cubic meter as 1 freight ton. If the cargo is less than 1 freight ton, it will be charged at 1 freight ton. By dividing a standard container into $\mathrm{{\rm Y}}$ shares ($\mathrm{{\rm Y}}$ = one standard container/freight ton), we decompose the FCL CFIM into the LCL CFIM.

In brief, the proposed CFIM meets the actual needs of SMIEEs. The design of the forward-starting term structure is consistent with the practice of SMIEEs, and the layered compensation structure satisfies the demand for cost transfer. In addition, the subdivision of containers suits the fragmentation characteristics of SMIEEs.

4.4. Pricing model

Based on the term, compensation and share structures, we derive the pricing model of the CFIM with and without compensation a cap. As index insurances have efficient purchase and claim settlement and very low management and transaction costs (Arandara et al., 2019, Assa and Wang, 2021), we focus on calculating the net premium per basic share of CFIM and ignore costs including sale and management expenses, shareholder returns and tax costs.

4.4.1. Net premium of CFIM without compensation cap

The net premium per basic share of the CFIM without cap, denoted $C_a(X)$, can be expressed as follows:

$$\begin{array}{c}{C}_a(X)=\mathrm{{\rm Y}}{e}^{-rT}\max {\{\theta (J-X),0\}}_{[0,T]}\\ =E{\left(\mathrm{{\rm Y}}\theta e^{-rT}\left(\left\{\begin{array}{c}\begin{array}{ccc}J-X,& \mathit{if}& J⩾X\end{array}\\ \begin{array}{ccc}0,& \mathit{if}& J⩽X\end{array}\end{array}\right)\right)\right)}_{[0,T]},\end{array}$$ (1)

where the settlement point of the CFIM contract is the arithmetic average of the container freight index during the valuation period, i.e., $J={\sum }_{t=t_1}^T{S}_t/\left(T-t_1\right)$ with $S_t$ being the point of the container freight index at time $t$ and $r$ being the pricing interest rate; $X$ denotes the trigger target index point; $\theta$ denotes the currency conversion coefficient of the freight index.

In terms of the specific pricing model of the CFIM, we choose the derivative pricing method. Actuarial pricing and derivative pricing are the two main methods for index insurance pricing, and the choice is normally based on considerations of the subject law, insurance structure, risk control requirements, and regulatory standards. For example, actuarial methods such as the classical Burn model are often used for the pricing of weather index insurance with typical periodicity (Han, et al., 2019). For the pricing of the CFIM, we adopt the derivative pricing for the following reasons: The stable matching between supply and demand of the global container shipping market and the asymmetry of shipping information leads to large and unpredictable volatilities of the container freight index (Jeon, et al., 2021). Therefore, the derivative pricing based on second moments can be more suitable for quantifying the risk under the insurance scenario. Also, the complex product structure is more consistent with the existing exotic derivatives structure. More specifically, under the forward-starting term structure, layered compensation structure and arithmetic mean settlement point, the CFIM is precisely the forward-starting arithmetic average Asian call option.

Following the approximate analytical formula of forward-starting arithmetic average Asian call option pricing proposed by Turnbull and Wakeman (1991), the pricing model of the CFIM without a compensation cap can be rewritten as follows:

$$C_a(X)\approx \mathrm{{\rm Y}}\theta \left[S_0{e}^{-rT}N(d_1)-X{e}^{-rT}N(d_2)\right],$$ (2)

$$d_1=\frac{\ln (S_0/X)+{\sigma }_A^{2}T\text{/2}}{{\sigma }_A\sqrt{T}},d_2=d_1-{\sigma }_A\sqrt{T},$$ (3)

$${\sigma }_A=\sqrt{\frac{\ln (M_1)}{T}},$$ (4)

$$M_1=\frac{2e^{{\sigma }^{2}T}-2e^{{\sigma }^2{t}_1}\left[1+{\sigma }^2(T-t_1)\right]}{{\sigma }^4(T-t_1)},$$ (5)

where $\sigma$ and ${\sigma }_A$ are the volatility and adjusted and annualized volatility of the container freight index, and $M_1$ is the adjusted and annualized volatility of the container freight index. $S_0$ is the initial point of the container freight index at the time $t=0$; $N(d_1)$ and $N(d_2)$ denote the cumulative distribution function of the standard normal distribution valued at $d_1$ and $d_2$, respectively.

It is worth noting that the volatility parameter plays a key role in pricing. Setting larger pricing volatilities can lead to higher net premiums, which is beneficial for insurers but less attractive for SMIEEs. On the other hand, lower net premiums due to setting smaller volatilities can attract more SMIEEs but will be less welcomed by insurers. Therefore, sufficient consideration should be given to the choice the volatility parameter.

4.4.2. Net premium of CFIM with compensation cap

The basic share of the CFIM with compensation cap, denoted $C_b(\mathrm{{\rm Y}}G+X)$, belongs to the interval compensation insurance, and the pricing model can be expressed as

$$C_b(\mathrm{{\rm Y}}G+X)=E{\left(\mathrm{{\rm Y}}\theta e^{-rT}\left(\left\{\begin{array}{ccc}G,& \mathit{if}& J⩾\mathrm{{\rm Y}}G+X\\ J-X,& \mathit{if}& \mathrm{{\rm Y}}G+X⩾J⩾X\\ 0,& \mathit{if}& J⩽X\end{array}\right)\right)\right)}_{[0,T]},$$ (6)

where $G$ is the insured amount and is the ceiling of indemnity, and other parameters are defined above. Following the approximation proposed by Turnbull and Wakeman (1991), Eq. (6) can be further derived as follows:

$$\begin{array}{ll}& C_b(\mathrm{{\rm Y}}G+X)\\ =& E{\left(\mathrm{{\rm Y}}\theta e^{-rT}\left(\left\{\begin{array}{ccc}G,& if& J⩾\mathrm{{\rm Y}}G+X\\ J-X,& if& \mathrm{{\rm Y}}G+X⩾J⩾X\\ 0,& if& J<X\end{array}\right)\right)\right)}_{[0,T]}\\ =& E{\left(\mathrm{{\rm Y}}\theta e^{-rT}\left(\left\{\begin{array}{ccc}J-X,& if& J⩾X\\ 0,& if& J<X\end{array}\right)\right)\right)}_{[0,T]}-E{\left(\mathrm{{\rm Y}}\theta e^{-rT}\left(\left\{\begin{array}{ccc}J-(\mathrm{{\rm Y}}G+X),& if& J⩾\mathrm{{\rm Y}}G+X\\ 0,& if& J<\mathrm{{\rm Y}}G+X\end{array}\right)\right)\right)}_{[0,T]}\\ =& C_a(X)-C_a(\mathrm{{\rm Y}}G+X)\\ =& \mathrm{{\rm Y}}\theta \left\{\left[S_0{e}^{-rT}N(d_1)-X{e}^{-rT}N(d_2)\right]-\left[S_0{e}^{-rT}N(d_1^{\ast })-X{e}^{-rT}N(d_2^{\ast })\right]\right\}.\end{array}$$ (7)

with $d_1^{\ast }$ and $d_2^{\ast }$ defined as

$$d_1^{*}=\frac{\ln (S_0/(\mathrm{{\rm Y}}G+X))+{\sigma }_A^{2}T\text{/2}}{{\sigma }_A\sqrt{T}},d_2^{*}=d_1^{*}-{\sigma }_A\sqrt{T}.$$ (8)

Notably, Eq. (7) becomes Eq. (2) when the insured indemnity ceiling tends to infinity, which implies that the CFIM without a cap can be viewed as a limiting case of the CFIM with a cap.

4.5. Scheme selection

For the principles of reliable index and adaptive structure, we have fully explained their roles in determining the underlying index and specific structure of CFIM. In the following, we further use the principles of affordable price and controllable risk to design relevant selection criteria so as to screen out the optimization scheme of the CFIM products.

The main challenge in designing the CFIM is to follow the principles of affordable price and controllable risk, which consider the interests of both insurers and SMIEEs. To achieve this target, we propose a set of selection criteria to choose the optimal scheme from the initial set of product schemes under various target points, extensions of pricing periods and insurance amounts.

More specifically, we use $\text{{X(}i\text{) + F(}j\text{) + G(}l\text{)}}$ to denote the initial set of product schemes, where $\text{X(}i\text{)}$ is the target point triggering compensation of the freight index in grade $i$, $\text{F(}j\text{)}$ is the settlement point for determining whether to settle claims or not when the valuation period forward-starting is $j$, and $\text{G(}l\text{)}$ is the insured indemnity ceiling in grade $l$. For example, in the case of insurance term $T=8$, F(0) is equivalent to no extension, and only the freight index point of the eighth period of the pricing period is used as the settlement point. In contrast, F(2) is equivalent to forward-starting $j=2$, and the arithmetic average of the freight index of the 6th, 7th and 8th valuation periods is taken as the claim settlement point. Without compensation cap, denoted $\text{G(U)}$, can be seen as a special case of the compensation cap. $\text{X(}i\text{)}$ and $\text{G(}l\text{)}$ should be set by insurers and SMIEEs through negotiations according to the fluctuation range and probability of historical data of the anchored container freight index.

4.5.1. Criteria for affordable price

Many criteria have been proposed as the measure of affordable prices, such as the basic share, insurance premium rate, and the number of covered groups (Institute of International Finance, 2018), which can be used for quantifying the affordability of the CFIM from different dimensions. As the basic share is already reflected in the product structure and the number of covered groups measures business results rather than product design, the insurance premium rate best reflects affordability. In practice, insurers generally set an upper bound for index or price rates, such as 10%, above which the premium rate will be so high that the leverage effect will be greatly reduced and the product will not be welcomed by SMIEEs.

We adopt the ANPR per basic share in the historical period as follows. Assuming that the historical period can be divided into $m$ periods, the average net premium (ANP), $P_m^{\prime }$, and ANPR, $P{R}_m^{\prime }$, for the basic share of the CFIM shall meet the following conditions:

$$P_m^{\prime }=\sum _{t=0}^m{C}_{b,t}(\mathrm{{\rm Y}}G+X)/(m+1),$$ (9)

$$P{R}_m^{\prime }=P_m^{\prime }/G⩽\varphi ,$$ (10)

where $C_{b,t}$ is the net premium per basic share for week $t$ of the historical period, and $\varphi$ is the highest acceptable rate level. For CFIM without a compensation cap, the condition is obviously satisfied since $G$ tends to infinity.

4.5.2. Criteria for controllable risk

The principle of controllable risk is crucial for insurers because it closely relates to the profitability and sustainability of the insurance business. In the practice of risk management, criteria including risk exposure, underwriting profit rate, comprehensive loss rate, and comprehensive cost rate are conventionally adopted. Risk exposure is mainly used for assessing the business progress rather than product design. Underwriting profit rate, comprehensive loss ratio and comprehensive cost ratio are measures of financial positions, among which underwriting profit rate is more representative. Therefore, we propose the measure of CUPR as elaborated below.

In practice, insurers set upper bounds on the total number of basic insured shares in a certain period to control the total insured amount. If the upper bound is reached, the insurer will stop selling. Continuous increase in container freights will motivate expectations for further increases and hence stimulate insurance demands; on the contrary, if the container freight continues to fall, it will lead to a reduced insurance amount. Therefore, we assume that the ratio $D_t$ of the underwriting amount of basic shares in period $t$ to the upper bound of the total amount of basic shares in each period is consistent with the change in the container freight index, which satisfies the condition:

$$D_t=\frac{S_t-\min {\left\{S_t\right\}}_{\mathrm{\Omega }}}{\mathit{max}{\left\{S_t\right\}}_{\mathrm{\Omega }}-\min {\left\{S_t\right\}}_{\mathrm{\Omega }}},$$ (11)

where ${\{S_t\}}_{\mathrm{\Omega }}$ is the data set of the container freight index during the retrospective period. For any of the $\text{X(}i\text{) + F(}j\text{) + G(}l\text{)}$ product schemes, the actual underwriting profit, denoted $R_{b,t}^{\prime }$, and CUPR, denoted $P{f}_m^{\prime }$, in the retrospective period time $t$ are calculated as follows:

$$R_{b,t}^{\prime }=D_t{C}_{b,t}-D_t{L}_{b,t},$$ (12)

$$P{f}_m^{\prime }=\sum _{t=0}^m{R}_{b,t}^{\prime }/\sum _{t=0}^m(D_t{C}_{b,t}),$$ (13)

wherein $L_{b,t}$ is the actual loss per basic share of CFIM with the period determined by Eqs. (2), (7). The CUPR shall meet the following condition

$$\text{0}⩽P{f}_m^{\prime }⩽\upsilon ,$$ (14)

where, $\upsilon$ is the upper bound of the CUPR in the retrospective period. The condition indicates that the CUPR cannot be less than 0 or higher than its upper bound in the retrospective period. It provides insurers with profit protection against extreme fluctuations in the container freight market.

5. Case study

With extraordinary capacities of liner containers, the CCFI Europe Service (CCFIES) is one of the busiest container shipping lines in the world, accounting for nearly €600 billion of annual merchandise trade. Therefore, we illustrate the design and scheme optimization process of the CFIM with the CCFIES index.

5.1. Data

The data covers the period from Mar. 14, 2003, to Aug. 13, 2021, consisting of 935 weeks, as shown in Fig. 3 . Data were collected from the iFinD database provided by Tong Hua Shun, a major financial data service company in China. The time interval from Mar. 14, 2003, to Dec. 25, 2020, (902 weeks) is taken as the retrospective period, and the time interval from Jan. 1, 2021, to Aug. 13, 2021, (33 weeks) is used to find the optimal schemes (shown as the pink area in Fig. 3). Due to the impact of the COVID-19 pandemic, the CCFIES index rose from 2318.13 on Jan. 1 to 5165.62 on Aug. 13, 2021, the period of which is very suitable for illustrating the optimization procedure under extreme scenarios. We chose the data for two reasons: First, the CCFI and its sub-line index were compiled by Shanghai Shipping Exchange in 1998. After the implementation of the new and improved data collection rules on Mar. 14, 2003, the index was gradually accepted by the global container shipping market and became the benchmark and authoritative container freight index in the world, which satisfies the reliable index principle of our product design. Second, product design and optimization scheme screening can effectively demonstrate their robustness, after a long period of inspection of the complete shipping cycle of 5–15 years (Stopford, 2008). In particular, the robustness of the product design and the selection procedure for the optimal product schemes can be tested in the scenario of severe fluctuation of freights in 2020.

Fig. 3.

The CCFIES index.

Table 1 exhibits the descriptive statistics. As can be seen, the weekly volatility of the CCFIES index is 2.98%, and the corresponding annualized volatility is 21.49% ($\text{2.98\%}\times \sqrt{52}$), which indicates that the CCFIES index is quite volatile.

Table 1.

Descriptive statistics of the CCFIES index (from Mar. 14, 2003, to Aug. 13, 2021).

	Max.	Min.	Mean	Std. Dev.	Skewness	Kurtosis
CCFIES Index	5230.11	625.12	1365.34	536.53	3.41	17.15
Return	22.19%	−9.39%	0.20%	2.98%	1.58	7.77

Note: Std. Dev. stand for Standard Deviation.

5.2. Parameters setting

The initial product scheme set of the basic share CCFIES index microinsurance, $\text{{ X(}i\text{) + F(}j\text{) + G(}l\text{)}}$, is designed as follows:

• (1)Set the insurance term to 8 weeks. In the CCFIES, the term of the container booking activities for SMIEEs is generally 8 weeks or 2 months.
• (2)Set the settlement point to 8 levels. The forward-starting term $j$ of the valuation period is divided into 8 weeks, which corresponds to the last $(j+1)$ week for $j=0,1,\cdots ,7$, respectively. The corresponding claim settlement points are the arithmetic mean of the weekly index points of the last $(j+1)$, denoted as $\text{F(}j\text{)}$.
• (3)Set the target index point triggering compensation into 5 grades, i.e.,$\text{X(}i\text{) (}i\text{= 0, 50, 100, 150, 200)}$, based on the CCFIES of the current week. More specifically, X(0) is the trigger point of the compensation, which is the CCFIES index point of the current week, X(50) is the trigger point of the CCFIES index of the current week plus 50 points, and $\text{X(}i\text{) (}i\text{= 100, 150, 200)}$ are defined analogously. The grades are determined according to the following procedure: Firstly, the CCFIES index settlement point minus the target points from the settlement points between Mar. 14, 2003, and Dec. 25, 2020, to obtain the corresponding weekly change points $\mathrm{\Delta }\text{F}(j)$, i.e., $\text{F}(j)-\text{X}(0)=\mathrm{\Delta }\text{F}(j)$. Secondly, the $\mathrm{\Delta }\text{F}(j)$ sequence is sorted from small to large, and the points of change are taken according to the percentile values of 50%, 75%, 85%, 90%, and 95%. Thirdly, the arithmetic average of $\mathrm{\Delta }\text{F}(j)$ is calculated. Taking 50 points as the basic unit, the trigger point $\text{X(}i\text{)}$ in 5 grades is determined following the nearest neighbor principle. It can be seen from Table 2 that the 200-point target has covered 95% of the historical uptrend, which indicates that the setting of 5 grades is appropriate.
• (4)Set the basic share $\mathrm{{\rm Y}}=10$. The CCFIES line mainly takes the TEU, which can be divided into 20 freight tons. Following the industrial rule that the minimum LCL booking is 2 freight tons, we take 10 as the basic share.
• (5)Set the insured amount per basic share $\text{G(}l\text{)}$. The $\text{G(}l\text{)}$ with compensation cap is set through the following two steps. First, we obtain the arithmetic average of the weekly change point values $\mathrm{\Delta }\text{F}(j)$, which corresponds to the 96%, 97%, and 98% percentiles of the historical period following the method of setting the trigger compensation, which leads to the gear point 250, 300, and 350 as shown in Table 2. Then, we set the insured amount by quantifying the risk of the freight index settlement point breaking through the trigger compensation $\text{X(}i\text{)}$ above. To be more precise, for example, the highest target point triggering compensation X(200), continues to move 250–200 = 50, 300–200 = 100 and 350–200 = 150 points higher, again covering 96%, 97% and 98% of historical weekly change point values $\mathrm{\Delta }\text{F}(j)$, respectively. Correspondingly, the insured amount per basic share can be set into 3 levels of 50/10 = 5, 100/10 = 10 and 150/10 = 15 points, namely, $\text{G(}l\text{) (}l\text{= 5, 10, 15)}$. Similarly, for $\text{X(}i\text{) (}i\text{= 50, 100, 150)}$, $\text{G(}l\text{) (}l\text{= 5, 10, 15)}$ ensures that 95% or more of the historical uptrend is covered. Specially, for X(0), we add the gear point 200, ensuring that it covered 95% of the historical uptrend. Consequently, the insured amount at the trigger point of the compensation is set into 4 grades, i.e.,$\text{G(}l\text{) (}l\text{= 5, 10, 15, 20)}$. In addition, the insured amount without compensation cap is recorded as $\text{G(U)}$.
• (6)Set the point value currency conversion coefficient to $1 per point, i.e., $\theta =1$.

Table 2.

Summary of the changes in target points.

Percentile	$\mathrm{\Delta }\text{F}(0)$	$\mathrm{\Delta }\text{F}(1)$	$\mathrm{\Delta }\text{F}(2)$	$\mathrm{\Delta }\text{F}(3)$	$\mathrm{\Delta }\text{F}(4)$	$\mathrm{\Delta }\text{F}(5)$	$\mathrm{\Delta }\text{F}(6)$	$\mathrm{\Delta }\text{F}(7)$	Mean $\left(\mathrm{\Delta }\text{F}(j)\right)$	Target point
50%	2	2	2	2	1	1	−1	1	1	0
75%	57	54	50	49	45	41	38	36	46	50
85%	106	101	95	91	82	77	70	65	86	100
90%	154	147	142	136	121	114	104	95	127	150
95%	253	234	212	207	189	182	150	153	197	200
96%	307	288	261	241	219	197	188	166	233	250
97%	364	319	311	289	272	254	215	197	278	300
98%	398	381	382	353	318	293	267	250	330	350

Table 3 shows the details of the initial product scheme set of the CCFIES index microinsurance, with a total of 176 product schemes.

Table 3.

The initial product scheme set of the CCFIES index microinsurance.

Component	Setting
Underlying index	The CCFIES index
Insurance period $T$	8 weeks (2 months)
Target point $\text{X(}i\text{)}$	5 levels, $i=0,50,100,150,200$ points up, based on the CCFIES for the week
Settlement point $\text{F(}j\text{)}$	The arithmetic average value of the weekly index points of last $(j+1)$ week, for $j=0,1,\cdots ,7$
Point value currency conversion coefficient $\theta$	$1 per point or equivalently $\theta =1$
Insured amount $\text{G(}l\text{)}$	For X(0), $\text{G(}l\text{)}$ is divided into 4 grades, with 5, 10, 15, and 20 points; for $\text{X(}i\text{) (}i\text{= 50, 100, 150, 200)}$ are divided into 3 grades, with 5, 10 and 15 points. Without compensation cap, $\text{G(U)}$.

The settings of the other relevant parameters are described as follows:

• (1)Volatility. The pricing volatility, $\sigma$, is set based on the CCFIES index in the retrospective period. First, the initial and annualized volatility series are calculated based on historical volatility with a window of 8 weeks, which is consistent with the insurance term of the CFIM. Second, the quantiles at the level of 95% and 99% are taken as the pricing volatility $\sigma$. Third, the annualized volatility series ${\sigma }_A$ of the different valuation periods can be obtained via Eq. (4). Fig. 4 shows the volatility trend at the 99% level. The volatility trend at the 95% level presents a similar pattern.
• (2)Pricing interest rate. Since the insurance term is very short, we assume a fixed annualized interest rate of $r=3\%$.
• (3)Premium rate cap. Following practice, the upper bound of the ANPR for all product schemes is set to be 10%, leading to an insurance leverage ratio of 1/10% = 10. This guarantees that the premium rates are not too high to be acceptable for SMIEEs, while insurers have sufficient risk control for sustainable operation.
• (4)Underwriting profit ratio cap. The retrospective period covers a span of nearly 18 years. Considering the current property insurance market in China, it seems reasonable to limit the average annual premium underwriting profit ratio of the CFIM to 2%. Then, by Eq. (14), the CUPR is capped at 36% while satisfying $P{f}_m^{\prime }>0$. Otherwise, the insurer’s long-term loss will not be in accordance with commercial insurance essentials.

Fig. 4.

8-weekly CCFIES index adjusted annual volatility at the level of 99%.

5.3. Premium calculation

The net premium per basic share of the CCFIES index microinsurance is calculated via Eqs. (2), (7). Table 4, Table 5 show the net premium per basic share with the descriptive statistics for all product schemes under the volatility levels of 99% and 95%, respectively. As can be seen from Table 4, Table 5, more conservative pricing volatility leads to higher premiums regardless of the setting of the compensation cap. Also, under the same conditions, the product scheme with a compensation cap has a lower premium than the product scheme without a compensation cap. Fig. 5 shows the net premiums per basic share for the product schemes of $\text{{X(50) + F(0, 1, 2, 3, 4, 5, 6, 7) + G(10, 15)}}$ at volatility level 99%. As can be seen, the premium has large variations between $0 and $2.50.

Table 4.

Weekly ANP per basic share at a volatility level of 99%.

Schemes	ANP	Max.	Min.	Std. Dev.	Schemes	ANP	Max.	Min.	Std. Dev.	Schemes	ANP	Max.	Min.	Std. Dev.
X(0) + F(0) + G(U)	3.45	57.50	16.80	7.95	X(0) + F(0) + G(5)	1.85	20.90	14.00	1.32	X(50) + F(0) + G(10)	1.35	24.20	2.79	4.58
X(0) + F(1) + G(U)	3.31	55.10	16.10	7.61	X(0) + F(1) + G(5)	1.83	20.70	13.70	1.36	X(50) + F(1) + G(10)	1.27	23.40	2.42	4.50
X(0) + F(2) + G(U)	3.15	52.50	15.40	7.25	X(0) + F(2) + G(5)	1.80	20.60	13.30	1.41	X(50) + F(2) + G(10)	1.18	22.50	2.05	4.38
X(0) + F(3) + G(U)	2.99	49.80	14.60	6.88	X(0) + F(3) + G(5)	1.78	20.40	12.90	1.47	X(50) + F(3) + G(10)	1.08	21.40	1.69	4.24
X(0) + F(4) + G(U)	2.82	47.00	13.70	6.49	X(0) + F(4) + G(5)	1.74	20.20	12.40	1.53	X(50) + F(4) + G(10)	0.98	20.20	1.34	4.05
X(0) + F(5) + G(U)	2.64	43.90	12.90	6.07	X(0) + F(5) + G(5)	1.71	20.00	11.80	1.60	X(50) + F(5) + G(10)	0.86	18.90	1.01	3.81
X(0) + F(6) + G(U)	2.44	40.70	11.90	5.62	X(0) + F(6) + G(5)	1.66	19.70	11.20	1.68	X(50) + F(6) + G(10)	0.74	17.20	0.71	3.51
X(0) + F(7) + G(U)	2.23	37.10	10.90	5.13	X(0) + F(7) + G(5)	1.60	19.30	10.40	1.77	X(50) + F(7) + G(10)	0.61	15.30	0.45	3.12
X(50) + F(0) + G(U)	1.61	36.70	2.80	6.67	X(0) + F(0) + G(10)	2.79	35.50	16.60	3.84	X(50) + F(0) + G(15)	1.51	30.10	2.80	5.76
X(50) + F(1) + G(U)	1.48	34.30	2.42	6.29	X(0) + F(1) + G(10)	2.72	35.00	15.90	3.89	X(50) + F(1) + G(15)	1.41	28.80	2.42	5.56
X(50) + F(2) + G(U)	1.35	31.90	2.05	5.89	X(0) + F(2) + G(10)	2.65	34.50	15.20	3.94	X(50) + F(2) + G(15)	1.29	27.30	2.05	5.31
X(50) + F(3) + G(U)	1.21	29.40	1.69	5.46	X(0) + F(3) + G(10)	2.57	33.90	14.50	3.98	X(50) + F(3) + G(15)	1.18	25.70	1.69	5.03
X(50) + F(4) + G(U)	1.08	26.70	1.34	5.01	X(0) + F(4) + G(10)	2.47	33.10	13.70	4.01	X(50) + F(4) + G(15)	1.05	24.00	1.34	4.71
X(50) + F(5) + G(U)	0.93	23.90	1.01	4.52	X(0) + F(5) + G(10)	2.36	32.30	12.80	4.03	X(50) + F(5) + G(15)	0.92	22.00	1.01	4.32
X(50) + F(6) + G(U)	0.79	21.00	0.71	3.99	X(0) + F(6) + G(10)	2.24	31.20	11.90	4.02	X(50) + F(6) + G(15)	0.78	19.70	0.71	3.88
X(50) + F(7) + G(U)	0.63	17.80	0.45	3.41	X(0) + F(7) + G(10)	2.09	29.90	10.90	3.96	X(50) + F(7) + G(15)	0.63	17.10	0.45	3.36
X(100) + F(0) + G(U)	0.67	22.00	0.23	4.20	X(0) + F(0) + G(15)	3.20	45.10	16.80	5.89	X(100) + F(0) + G(5)	0.41	9.58	0.22	2.07
X(100) + F(1) + G(U)	0.59	20.00	0.17	3.81	X(0) + F(1) + G(15)	3.09	44.20	16.10	5.84	X(100) + F(1) + G(5)	0.37	9.11	0.16	1.97
X(100) + F(2) + G(U)	0.51	18.00	0.11	3.41	X(0) + F(2) + G(15)	2.98	43.10	15.40	5.78	X(100) + F(2) + G(5)	0.33	8.59	0.11	1.86
X(100) + F(3) + G(U)	0.43	15.90	0.07	2.99	X(0) + F(3) + G(15)	2.86	41.90	14.60	5.68	X(100) + F(3) + G(5)	0.29	8.00	0.07	1.73
X(100) + F(4) + G(U)	0.35	13.80	0.04	2.57	X(0) + F(4) + G(15)	2.72	40.50	13.70	5.56	X(100) + F(4) + G(5)	0.25	7.34	0.04	1.58
X(100) + F(5) + G(U)	0.27	11.70	0.02	2.13	X(0) + F(5) + G(15)	2.57	38.90	12.90	5.38	X(100) + F(5) + G(5)	0.20	6.59	0.02	1.39
X(100) + F(6) + G(U)	0.20	9.48	0.01	1.69	X(0) + F(6) + G(15)	2.40	36.90	11.90	5.16	X(100) + F(6) + G(5)	0.16	5.74	0.01	1.18
X(100) + F(7) + G(U)	0.14	7.28	0.00	1.24	X(0) + F(7) + G(15)	2.20	34.60	10.90	4.85	X(100) + F(7) + G(5)	0.11	4.76	0.00	0.93
X(150) + F(0) + G(U)	0.26	12.40	0.01	2.17	X(0) + F(0) + G(20)	3.36	50.90	16.80	7.05	X(100) + F(0) + G(10)	0.58	15.40	0.23	3.26
X(150) + F(1) + G(U)	0.21	10.90	0.01	1.87	X(0) + F(1) + G(20)	3.24	49.50	16.10	6.89	X(100) + F(1) + G(10)	0.51	14.50	0.17	3.05
X(150) + F(2) + G(U)	0.17	9.41	0.00	1.58	X(0) + F(2) + G(20)	3.10	47.90	15.40	6.69	X(100) + F(2) + G(10)	0.45	13.40	0.11	2.81
X(150) + F(3) + G(U)	0.13	7.93	0.00	1.29	X(0) + F(3) + G(20)	2.95	46.20	14.60	6.47	X(100) + F(3) + G(10)	0.39	12.30	0.07	2.55
X(150) + F(4) + G(U)	0.10	6.48	0.00	1.02	X(0) + F(4) + G(20)	2.79	44.20	13.70	6.20	X(100) + F(4) + G(10)	0.32	11.10	0.04	2.25
X(150) + F(5) + G(U)	0.07	5.08	0.00	0.76	X(0) + F(5) + G(20)	2.62	42.00	12.90	5.88	X(100) + F(5) + G(10)	0.26	9.70	0.02	1.92
X(150) + F(6) + G(U)	0.04	3.74	0.00	0.52	X(0) + F(6) + G(20)	2.43	39.40	11.90	5.51	X(100) + F(6) + G(10)	0.20	8.19	0.01	1.56
X(150) + F(7) + G(U)	0.03	2.52	0.00	0.32	X(0) + F(7) + G(20)	2.23	36.40	10.90	5.08	X(100) + F(7) + G(10)	0.14	6.54	0.00	1.18
X(200) + F(0) + G(U)	0.09	6.58	0.00	0.98	X(50) + F(0) + G(5)	0.94	14.70	2.57	2.53	X(150) + F(0) + G(5)	0.16	5.83	0.01	1.21
X(200) + F(1) + G(U)	0.07	5.55	0.00	0.80	X(50) + F(1) + G(5)	0.89	14.30	2.26	2.54	X(150) + F(1) + G(5)	0.14	5.36	0.01	1.09
X(200) + F(2) + G(U)	0.05	4.56	0.00	0.63	X(50) + F(2) + G(5)	0.84	13.90	1.94	2.54	X(150) + F(2) + G(5)	0.12	4.85	0.00	0.96
X(200) + F(3) + G(U)	0.04	3.62	0.00	0.48	X(50) + F(3) + G(5)	0.79	13.40	1.62	2.52	X(150) + F(3) + G(5)	0.10	4.31	0.00	0.83
X(200) + F(4) + G(U)	0.03	2.75	0.00	0.34	X(50) + F(4) + G(5)	0.73	12.90	1.30	2.49	X(150) + F(4) + G(5)	0.07	3.73	0.00	0.68
X(200) + F(5) + G(U)	0.02	1.97	0.00	0.22	X(50) + F(5) + G(5)	0.66	12.30	0.99	2.44	X(150) + F(5) + G(5)	0.05	3.11	0.00	0.54
X(200) + F(6) + G(U)	0.01	1.29	0.00	0.13	X(50) + F(6) + G(5)	0.58	11.50	0.70	2.35	X(150) + F(6) + G(5)	0.04	2.45	0.00	0.39
X(200) + F(7) + G(U)	0.00	0.73	0.00	0.07	X(50) + F(7) + G(5)	0.49	10.50	0.45	2.21	X(150) + F(7) + G(5)	0.02	1.78	0.00	0.26

Table 5.

Weekly ANP per basic share at volatility level of 95%.

Schemes	ANP	Max.	Min.	Std. Dev.	Schemes	ANP	Max.	Min.	Std. Dev.	Schemes	ANP	Max.	Min.	Std. Dev.
X(0) + F(0) + G(U)	2.65	44.10	12.90	6.10	X(0) + F(0) + G(5)	1.71	20.00	11.90	1.60	X(50) + F(0) + G(10)	0.87	19.00	1.03	3.83
X(0) + F(1) + G(U)	2.54	42.30	12.40	5.84	X(0) + F(1) + G(5)	1.68	19.90	11.50	1.64	X(50) + F(1) + G(10)	0.80	18.00	0.85	3.66
X(0) + F(2) + G(U)	2.42	40.30	11.80	5.57	X(0) + F(2) + G(5)	1.65	19.70	11.10	1.69	X(50) + F(2) + G(10)	0.73	17.00	0.68	3.47
X(0) + F(3) + G(U)	2.30	38.20	11.20	5.28	X(0) + F(3) + G(5)	1.62	19.50	10.70	1.74	X(50) + F(3) + G(10)	0.65	15.90	0.52	3.25
X(0) + F(4) + G(U)	2.17	36.00	10.50	4.98	X(0) + F(4) + G(5)	1.58	19.20	10.20	1.80	X(50) + F(4) + G(10)	0.57	14.70	0.38	2.99
X(0) + F(5) + G(U)	2.03	33.70	9.86	4.66	X(0) + F(5) + G(5)	1.53	18.90	9.60	1.86	X(50) + F(5) + G(10)	0.48	13.30	0.26	2.70
X(0) + F(6) + G(U)	1.88	31.20	9.13	4.31	X(0) + F(6) + G(5)	1.47	18.50	8.97	1.92	X(50) + F(6) + G(10)	0.40	11.70	0.16	2.36
X(0) + F(7) + G(U)	1.71	28.50	8.33	3.94	X(0) + F(7) + G(5)	1.40	18.00	8.25	1.98	X(50) + F(7) + G(10)	0.31	9.93	0.08	1.97
X(50) + F(0) + G(U)	0.94	24.10	1.03	4.55	X(0) + F(0) + G(10)	2.37	32.30	12.90	4.03	X(50) + F(0) + G(15)	0.93	22.10	1.03	4.35
X(50) + F(1) + G(U)	0.86	22.40	0.85	4.25	X(0) + F(1) + G(10)	2.30	31.70	12.40	4.03	X(50) + F(1) + G(15)	0.85	20.80	0.85	4.10
X(50) + F(2) + G(U)	0.77	20.60	0.68	3.93	X(0) + F(2) + G(10)	2.22	31.10	11.80	4.01	X(50) + F(2) + G(15)	0.76	19.40	0.68	3.82
X(50) + F(3) + G(U)	0.68	18.80	0.52	3.59	X(0) + F(3) + G(10)	2.14	30.30	11.20	3.98	X(50) + F(3) + G(15)	0.67	17.90	0.52	3.52
X(50) + F(4) + G(U)	0.59	16.90	0.38	3.23	X(0) + F(4) + G(10)	2.04	29.40	10.50	3.93	X(50) + F(4) + G(15)	0.59	16.30	0.38	3.19
X(50) + F(5) + G(U)	0.50	14.90	0.26	2.85	X(0) + F(5) + G(10)	1.94	28.40	9.86	3.85	X(50) + F(5) + G(15)	0.49	14.50	0.26	2.83
X(50) + F(6) + G(U)	0.40	12.70	0.16	2.45	X(0) + F(6) + G(10)	1.81	27.10	9.13	3.74	X(50) + F(6) + G(15)	0.40	12.50	0.16	2.44
X(50) + F(7) + G(U)	0.31	10.50	0.08	2.01	X(0) + F(7) + G(10)	1.67	25.60	8.33	3.56	X(50) + F(7) + G(15)	0.31	10.40	0.08	2.01
X(100) + F(0) + G(U)	0.28	11.80	0.02	2.16	X(0) + F(0) + G(15)	2.58	39.00	12.90	5.40	X(100) + F(0) + G(5)	0.21	6.65	0.02	1.41
X(100) + F(1) + G(U)	0.24	10.50	0.01	1.90	X(0) + F(1) + G(15)	2.48	37.90	12.40	5.27	X(100) + F(1) + G(5)	0.18	6.16	0.01	1.29
X(100) + F(2) + G(U)	0.20	9.24	0.01	1.64	X(0) + F(2) + G(15)	2.38	36.70	11.80	5.13	X(100) + F(2) + G(5)	0.15	5.64	0.01	1.15
X(100) + F(3) + G(U)	0.16	7.94	0.00	1.38	X(0) + F(3) + G(15)	2.27	35.40	11.20	4.95	X(100) + F(3) + G(5)	0.13	5.07	0.00	1.01
X(100) + F(4) + G(U)	0.12	6.65	0.00	1.12	X(0) + F(4) + G(15)	2.14	33.90	10.50	4.75	X(100) + F(4) + G(5)	0.10	4.46	0.00	0.86
X(100) + F(5) + G(U)	0.09	5.36	0.00	0.87	X(0) + F(5) + G(15)	2.01	32.10	9.86	4.51	X(100) + F(5) + G(5)	0.08	3.79	0.00	0.70
X(100) + F(6) + G(U)	0.06	4.11	0.00	0.63	X(0) + F(6) + G(15)	1.87	30.20	9.13	4.23	X(100) + F(6) + G(5)	0.05	3.08	0.00	0.53
X(100) + F(7) + G(U)	0.04	2.92	0.00	0.42	X(0) + F(7) + G(15)	1.71	27.90	8.33	3.90	X(100) + F(7) + G(5)	0.03	2.32	0.00	0.37
X(150) + F(0) + G(U)	0.07	5.17	0.00	0.77	X(0) + F(0) + G(20)	2.63	42.10	12.90	5.91	X(100) + F(0) + G(10)	0.26	9.80	0.02	1.95
X(150) + F(1) + G(U)	0.06	4.37	0.00	0.63	X(0) + F(1) + G(20)	2.53	40.70	12.40	5.70	X(100) + F(1) + G(10)	0.23	8.93	0.01	1.74
X(150) + F(2) + G(U)	0.04	3.60	0.00	0.50	X(0) + F(2) + G(20)	2.41	39.10	11.80	5.47	X(100) + F(2) + G(10)	0.19	8.02	0.01	1.52
X(150) + F(3) + G(U)	0.03	2.87	0.00	0.38	X(0) + F(3) + G(20)	2.29	37.30	11.20	5.22	X(100) + F(3) + G(10)	0.15	7.06	0.00	1.30
X(150) + F(4) + G(U)	0.02	2.19	0.00	0.27	X(0) + F(4) + G(20)	2.16	35.40	10.50	4.94	X(100) + F(4) + G(10)	0.12	6.04	0.00	1.07
X(150) + F(5) + G(U)	0.01	1.57	0.00	0.18	X(0) + F(5) + G(20)	2.02	33.30	9.86	4.64	X(100) + F(5) + G(10)	0.09	4.99	0.00	0.84
X(150) + F(6) + G(U)	0.01	1.03	0.00	0.11	X(0) + F(6) + G(20)	1.87	31.00	9.13	4.30	X(100) + F(6) + G(10)	0.06	3.91	0.00	0.62
X(150) + F(7) + G(U)	0.00	0.60	0.00	0.05	X(0) + F(7) + G(20)	1.71	28.40	8.33	3.93	X(100) + F(7) + G(10)	0.04	2.83	0.00	0.41
X(200) + F(0) + G(U)	0.02	2.02	0.00	0.23	X(50) + F(0) + G(5)	0.66	12.30	1.01	2.44	X(150) + F(0) + G(5)	0.06	3.15	0.00	0.55
X(200) + F(1) + G(U)	0.01	1.60	0.00	0.17	X(50) + F(1) + G(5)	0.62	11.90	0.84	2.40	X(150) + F(1) + G(5)	0.04	2.77	0.00	0.46
X(200) + F(2) + G(U)	0.01	1.22	0.00	0.12	X(50) + F(2) + G(5)	0.57	11.40	0.67	2.34	X(150) + F(2) + G(5)	0.03	2.38	0.00	0.38
X(200) + F(3) + G(U)	0.01	0.89	0.00	0.08	X(50) + F(3) + G(5)	0.52	10.80	0.52	2.26	X(150) + F(3) + G(5)	0.03	1.99	0.00	0.30
X(200) + F(4) + G(U)	0.00	0.60	0.00	0.05	X(50) + F(4) + G(5)	0.47	10.20	0.38	2.15	X(150) + F(4) + G(5)	0.02	1.59	0.00	0.22
X(200) + F(5) + G(U)	0.00	0.37	0.00	0.03	X(50) + F(5) + G(5)	0.41	9.49	0.26	2.02	X(150) + F(5) + G(5)	0.01	1.20	0.00	0.15
X(200) + F(6) + G(U)	0.00	0.20	0.00	0.01	X(50) + F(6) + G(5)	0.34	8.63	0.16	1.84	X(150) + F(6) + G(5)	0.01	0.83	0.00	0.09
X(200) + F(7) + G(U)	0.00	0.09	0.00	0.01	X(50) + F(7) + G(5)	0.27	7.61	0.08	1.62	X(150) + F(7) + G(5)	0.00	0.51	0.00	0.05

Fig. 5.

Weekly net premium per basic share of CCFIES index microinsurance at a volatility level of 99%.

5.4. Scheme selection

We adopt the ANPR and CUPR to find the optimal product scheme. The period from Mar. 14, 2003, to Dec. 25, 2020, is still selected. Fig. 6 shows the distribution curve of the total number of weekly underwriting basic shares of the CCFIES index microinsurance calculated via Eq. (11) for the retrospective period. It can be seen that the weekly underwriting volumes are in line with the changing trend of the CCFIES index.

Fig. 6.

Weekly coverage distribution of CCFIES index microinsurance during the retrospective period.

Fig. 7 shows the weekly underwriting profits of the CCFIES index microinsurance over the retrospective period calculated via Eq. (12) at the volatility level of 99%. As can be seen, the profit series presents typical patterns of large aggregate loss, which is particularly observable in the years 2007, 2009 and 2012. This means that the CCFIES index microinsurance requires earnings from other periods to cover the losses, which also highlights the importance of weekly underwriting caps for insurers.

Fig. 7.

Weekly underwriting profit of CCFIES index microinsurance at the volatility level of 99%.

Fig. 8, Fig. 9 show the ANP, ANPR and CUPR at the volatility level of 99% under the conditions of with and without a compensation cap. We can see that all product schemes without compensation cap suffer from loss and hence are dropped from the selection. In contrast, the product schemes with a compensation cap of $\text{{X(50) + F(0, 1, 2, 3, 4, 5, 6, 7) + G(5, 10, 15)}}$ and $\text{{X(100) + F(0, 1) + G(5)}}$ meet the requirement of all three criteria. Taking X(50) + F(6) + G(15) as an example, the ANP is 0.78, the ANPR is 5.16%, which is 10% below the upper bound, and the CUPR is 19.72%, which is in the 0–36% range.

Fig. 8.

ANP and CUPR without compensation cap at volatility level of 99%.

Fig. 9.

ANP, ANPR and CUPR with compensation cap at volatility level of 99%.

Analogously, we also calculate the criteria for all product schemes at the volatility level of 95%, as shown in Fig. 10, Fig. 11 . Again, the product schemes without a compensation cap all suffer loss. On the other hand, only the CUPRs of the product schemes of $\text{{X(0) + F(0, 1, 2, 3, 4, 5, 6, 7) + G(5)}}$ and $\text{{X(0) + F(0, 1, 2, 3, 4) + G(10)}}$ are greater than zero. Meanwhile, the ANPRs of these product schemes are more than 20%, which is obviously unattractive for SMIEEs. In brief, SMIEEs and insurers have more options under the setting of higher volatility levels.

Fig. 10.

ANP and CUPR without compensation cap at volatility level of 95%.

Fig. 11.

ANP, ANPR and CUPR with compensation cap at volatility level of 95 %.

Based on the criterion tests for the retrospective period, we select 12 product schemes at a volatility level of 99%, i.e.,$\text{{X(50) + F(4, 5, 6, 7) + G(10), X(50) + F(0, 1, 2, 3, 4, 5, 6, 7) + G(15)}}$, as summarized in Table 6 . These product schemes have good performance even in extreme scenarios, especially during the booms of the global container shipping market in 2007, 2009 and 2012. The selected product schemes are featured with relatively low ANPR, and the ANPR is no more than 10%, providing high insurance leverage for SMIEEs. Meanwhile, the CUPRs of these product schemes are more than 0 and less than 36%, which means insurers can effectively make cross-term and cross-enterprise risk diversifications.

Table 6.

The performance of selected product schemes in the retrospective period.

Schemes	Proposed model			Burn model
	ANP	ANPR	CUPR	ANP	ANPR	CUPR
X(50) + F(4) + G(10)	0.98	9.73%	34.23%	1.25	12.5 %	29.43%
X(50) + F(5) + G(10)	0.86	8.60%	32.20%	1.15	11.52%	29.53%
X(50) + F(6) + G(10)	0.74	7.38%	29.82%	1.00	10.04%	30.88%
X(50) + F(7) + G(10)	0.61	6.05%	25.65%	0.93	9.30%	30.04%
X(50) + F(0) + G(15)	1.51	10.00%	39.84%	1.95	12.99%	30.13%
X(50) + F(1) + G(15)	1.41	9.35%	38.54%	1.85	12.36%	30.29%
X(50) + F(2) + G(15)	1.29	8.61%	36.38%	1.75	11.68%	30.20%
X(50) + F(3) + G(15)	1.18	7.82%	33.37%	1.65	10.97%	30.11%
X(50) + F(4) + G(15)	1.05	6.98%	30.21%	1.53	10.18%	30.13%
X(50) + F(5) + G(15)	0.92	6.09%	25.25%	1.40	9.35%	30.13%
X(50) + F(6) + G(15)	0.78	5.16%	19.72%	1.21	8.04%	32.05%
X(50) + F(7) + G(15)	0.63	4.18%	9.79%	1.11	7.43%	30.59%

We revisit the YR order to see how insurance can make a difference. If YR purchased a 2-month term insurance and selected the X(50) + F(3) + G(15) product scheme On Oct.17, 2020. The ANP on the corresponding day was $0.31. So YR needed to pay a premium of 5 * 0.31 = $1.55. When the CFIM product was matured, YR would receive an insurance payout of $63.43. This reduced the loss of YR caused by freight increase.

5.5. Discussions

We use the classical Burn model (Han, et al., 2019) to analyze the selected product scheme of the CFIM. The Burn model assumes that the probability distribution of future losses is consistent with historical experience and takes the expectation based on historical data as the optimal estimate of the net premium. The pricing of the CFIM under the Burn model is as follows: We set the rolling window of periods to be 1 year (48 weeks). Then we obtain the new sequence of the change points, i.e., the difference between the settlement point and the target points, based on the CCFIES index for each time window. Finally, we calculate the average of the new sequence points that are larger than 0, which is the weekly net premium.

We use the Burn model to calculate the ANP, ANPR and CUPR per basic share for the selected product schemes, i.e.,$\text{{X(50) + F(4, 5, 6, 7) + G(10), X(50) + F(0, 1, 2, 3, 4, 5, 6, 7) + G(15)}}$, as shown in Table 6. As can be seen, under the proposed model, the insurance leverage of the proposed model is significantly higher. Furthermore, the ANPRs calculated by the proposed model are less than 10%, while those of the Burn model can be above 10%. Also, the CUPR decreases with the increase in the valuation period, while the change under the Burn model is not relatively small, within a level of approximately 30%.

We further test the robustness of the selected product schemes during the boom of the CCFIES index between Jan. 1 and Aug. 13, 2021. For consistency, we assume that the weekly insurance volume is based on the maximum of the total number of basic shares. The test results are shown in Table 7 . It can be seen that, during the testing period, the CUPRs of the selected product schemes drop by nearly 10% compared with the backdating historical period. In response to the rapid increase in the CCFIES index during the testing period, the calculation in our model shows that the ANPR and net premium also increase adaptively. In contrast, as shown in Table 7, although the ANPRs and ANPs under the Burn model also increase accordingly, the CUPR drops by more than 20%, which can be unacceptable for insurers.

Table 7.

The Performance of selected schemes in the testing period from Mar. 14, 2003, to Aug. 13, 2021.

Schemes	Proposed model			Burn model
	ANP	ANPR	CUPR	ANP	ANPR	CUPR
X(50) + F(4) + G(10)	1.05	10.47%	24.95%	1.36	13.59%	5.00%
X(50) + F(5) + G(10)	0.93	9.34%	21.55%	1.26	12.55%	2.95%
X(50) + F(6) + G(10)	0.81	8.11%	17.03%	1.11	11.12%	1.19%
X(50) + F(7) + G(10)	0.68	6.77%	7.23%	1.03	10.29%	−2.10%
X(50) + F(0) + G(15)	1.62	10.77%	39.99%	2.12	14.12%	8.19%
X(50) + F(1) + G(15)	1.51	10.06%	38.07%	2.03	13.50%	6.22%
X(50) + F(2) + G(15)	1.40	9.31%	35.24%	1.92	12.80%	4.60%
X(50) + F(3) + G(15)	1.28	8.52%	29.94%	1.81	12.06%	2.81%
X(50) + F(4) + G(15)	1.15	7.67%	26.07%	1.69	11.24%	0.81%
X(50) + F(5) + G(15)	1.02	6.77%	17.48%	1.56	10.38%	−1.52%
X(50) + F(6) + G(15)	0.87	5.82%	10.32%	1.37	9.14%	−3.37%
X(50) + F(7) + G(15)	0.72	4.80%	−3.16%	1.26	8.42%	−7.23%

To sum up, it is found that the forward-starting insurance with the target point, limited payouts and conservative pricing meets the needs of the CFIM. For example, under extreme conditions, the selected product scheme X(50) + F(6) + G(15), maintains reasonable ANPR (5.82%) and CUPR (10.32%).

The screened product solutions have important implications for relevant stakeholders. SMIEEs can make full use of the CFIM to transfer the cost of freight increase, especially under the scenario of rapid increases in container freight. In this process, SMIEEs need to determine the insurance period, basic share quantity and acceptable premium according to the trade order period, freight volume and their financial capacity. Then, a specific product will be selected from the specifically selected product scheme for insurance to lock in the additional or increased freight cost. Insurers also need to dynamically adjust these optimized product plans to ensure that the attractiveness of products and risks are controllable. In addition, CFIM is a kind of green box policy financial instrument to support import and export trade. The government departments of foreign trade or commerce could provide certain financial premium subsidies and other policy support to continuously expand the coverage of CFIM.

6. Conclusions

SMIEEs have long faced the challenge of additional costs caused by container freight fluctuations under fragmented transportation. The impact of COVID-19 has further added to the costs of fragmented space booking activities. In this study, we propose a design of the CFIM and illustrate the framework with the application to the CCFIES index. First, according to the market demand and inclusion, we put forward four basic principles framework for the design of index microinsurance, which is used as the benchmark for CFIM designs. Second, considering the characteristics of actual booking activities, we analyze the term, compensation and share structures of the insurance product, and design two basic share products of the CFIM without and with the compensation cap. Third, with reference to the forward-starting arithmetic average Asian call option pricing model, we establish the dynamic pricing models for the product under various settings, which allows dynamic calculation and adjustment of the net premium. Fourth, we further propose the selection procedure for the product schemes based on criteria including the ANPR and the CUPR to ensure the designed product meets the interests of both SMIEEs and insurers. Last, we conduct a case study of the CCFIES index microinsurance using 18-year retrospective period weekly data. The empirical results indicate that the CFIM can effectively meet SMIEEs’ needs for risk management and cost control.

This study is of potential interest to both academics and practitioners. The proposed RAAC principles (reliable index, adaptive structure, affordable price, and controllable risk) can shed light on the design of other index microinsurance. The dynamic pricing model derived for the index microinsurance with the compensation cap may also be useful for the pricing of freight options bilaterally. The CFIM designed in this paper can effectively help SMIEEs control the risk of additional costs caused by container freight fluctuations, especially under extreme market conditions such as boomed freights.

The Nymex Exchange of the CME Group launched the FBX container index futures on Feb. 28, 2022, China also has been preparing to re-launch the container freight index futures. The CFIM designed in this paper can be used for hedging with the aid of the container freight index futures, which can not only help insurers to increase underwriting capability but also provide SMIEEs with strong support for risk management. In this context, it is of interest to further explore the hedging strategies for the underwriting risk of CFIMs using the container freight index futures and the financial subsidies to CFIM premiums, so as to promote the wider application of the CFIM and the stable development of the global maritime supply chain.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary Data

The following are the Supplementary data to this article:

Data availability

Data provided in the supplementary material.