Efficiency comparison of selected OECD life insurance markets using a three-stage DEA

© 2021 Published by VŠB-TU Ostrava. All rights reserved. ER-CEREI, Volume 24: 55–68 (2021). ISSN 1212-3951 (Print), 1805-9481 (Online) Efficiency comparison of selected OECD life insurance markets using a three-stage DEA model Biwei GUAN a a Department of Finance, Faculty of Economics, VŠB – Technical University of Ostrava, Sokolská tř. 33, Ostrava 70200, Czech Republic. Abstract This paper compares the efficiency of 12 selected OECD life insurance markets from 2013 to 2019 through the three-stage DEA model, identifies ways to improve the efficiency of the potentially inefficient insurance markets, and determines how environmental factors influence the efficiency score. The major contribution of this paper is that, by using the three-stage DEA model and eliminating the impact of environmental factors on efficiency, the results are more accurate than those of previous studies. We find that the environmental factors have little effect on the German, Irish, and Italian life insurance markets, which perform well. However, after removing the influ-ence of environmental factors, the technical efficiency of the Belgian, Greek, and Hungarian markets decreases significantly. Keywords Panel data, three-stage DEA model, life-insurance market, OCED countries, efficiency measurement JEL Classification: C67, G15, G22 56 Ekonomická revue – Central European Review of Economic Issues 24, 2021 Efficiency comparison of selected OECD life insurance markets using a three-stage DEA model Biwei GUAN 1. Introduction and Literature Review In recent years, efficiency measurement in the insurance industry has become popular and has attracted the attention of many regulators and investors. Eling and Luhnen (2010) mentioned that, from 2000 to 2010, more than 90 studies focused on efficiency measurement in the insurance industry. Kaffash et al. (2019) pointed out that, between 1993 and 2018, 132 studies on the application of data envelopment analysis (DEA) in the insurance industry were pub-lished. Nowadays, the amount of research on this topic is continuing to grow. There are two main methods for the measurement of efficiency: stochastic frontier analysis (SFA) and data envelopment analysis (DEA). In previous studies, DEA and SFA have frequently been used to assess the efficiency of the insurance industry. In the beginning, SFA had an advantage over DEA in that it can ana-lyse the influencing factors. However, after many studies on improving and innovating the DEA model had been produced, such as those by Barros et al. (2010), Kao and Hwang (2014), and Yang and Pollitt (2009), this advantage weakened considerably. Kaffash et al. (2019) mentioned that, in recent studies on the insurance industry, DEA has been used more than SFA. Farrell (1957) introduced the basic DEA model to evaluate the efficiency of modern compa-nies; on this basis, Charnes et al. (1978) and Banker et al. (1984) introduced the CCR model and the BCC model, respectively. Later, some researchers pointed out that the traditional DEA model ignored the influ-ence of environmental effects and statistical noise on decision-making units (DMUs). Fried et al. (2002) proposed a threestage DEA model to eliminate the influence of the above two factors. As an international economic organization, the Organisation for Economic Co-operation and Devel-opment (OECD) currently includes 38 member states. The authors of a large number of previous studies have selected OECD insurance markets as their research object, although the types and numbers of the insurance markets selected have differed. Donni and Fecher (1997) investigated the life insurance and non-life insurance industries in 15 OECD countries; Davutyan and Klumpes (2008) surveyed the life and non-life insurance industries in seven OECD countries; Diacon (2001) applied the DEA model to analyse the technical efficiency (TE) of six OECD general insur-ance markets; and Diacon et al. (2002) studied the pure technical efficiency (PTE) and scale efficiency (SE) of 15 OECD life insurance markets through the DEA model. In this paper, we calculate the value of technical efficiency, pure technical efficiency, and scale efficiency of 12 selected OECD life insurance markets. To obtain the related efficiency value, we split the panel data into cross-sectional data, calculate the efficiency scores, respectively, and summarize them. Regarding the selected markets, we choose Belgium, Denmark, Finland, Germany, Greece, Hungary, Ireland, Italy, Luxemburg, Poland, Portugal, and Spain as the DMUs in this paper. As we mentioned earlier, the OECD has 38 member countries, located in different regions, such as Europe, America, and Asia. Their cultures, economies, and consumer habits are very different, which, to a certain extent, can lead to significant differences in the life insurance market. To minimize the gaps in the external environment of different life insurance markets, we first select EU countries (22 in total) in the OECD, which, to some extent, have stronger similarities, for example in terms of the regulation of the insurance industry. Unfortu-nately, however, when collecting the data, we find that some countries have incomplete data, such as Latvia, the Slovak Republic, and Slovenia with incomplete data on gross premiums, the Czech Re-public, France, and the Netherlands with missing data on total investments, and so on. To maintain the consistency of the data for the selected markets, we discard these countries with incomplete data and therefore end up with 12 life insurance markets as the research objectives of this paper. Eling and Luhnen (2010) calculated the average TE value of these selected life insurance industries as 0.73 (Belgium), 0.89 (Denmark), 0.84 (Finland), 0.79 (Germany), 0.70 (Ireland), 0.78 (Italy), 0.89 (Luxem-burg), 0.63 (Poland), 0.78 (Portugal), and 0.82 (Spain). It is worth mentioning that, in our previous research, the TE of the German life insurance market was 1 between 2013 and 2017, which is quite different from the results B. Guan – Efficiency comparison of selected OECD life insurance markets using a three-stage DEA model 57 of this paper. In this paper, we eliminate the impact of environmental effects and statistical noise to see whether we can obtain different answers. The purpose of this paper is to compare the efficiency of the 12 selected OECD life insurance markets from 2013 to 2019 through the three-stage DEA model, to find ways to improve the efficiency of the potentially inefficient insurance markets, and to determine how the environmental factors influence the efficiency score. The contribution of this paper is that most of the previous studies that selected OECD countries as their research objects used the basic or two-stage DEA model and did not consider the impact of environmental factors. This paper uses the latest data for the analysis and eliminates the impact of environmental factors. Thus, the results are more accurate. This paper is divided into five sections. In section 1, we start with a literature review of this topic, briefly introduce the issues, the current state of their resolution, and the purpose, structures, and main contribution of this paper. In section 2, we introduce the background information of the selected life insurance markets in detail. In section 3, we summarize the related data and explain why they were chosen. Then, we introduce the specific information on the three-stage DEA model. In section 4, the efficiency score of each insurance market is presented as well as a comparison of these results. We consider how to improve the efficiency score by decreasing the related input value and the influence of environmental factors on the efficiency score. In section 5, we summarize the paper, including its main contribution and key findings. 2. General Information of the Selected Life Insurance Industries Firstly, OECD member states have a relationship of mutual supervision and promotion. They have a closer relationship in many fields. Compared with many independent insurance markets, they have greater research value. Secondly, OECD member states include most of the countries that occupy a large market share of the global insurance market or an important position in the history of insurance development. Moreover, relevant data on the OECD insurance markets are easy to collect and accurate. As we mentioned earlier, in the traditional DEA model, the efficiency score is affected by environ-mental factors and statistical noise. Although the 12 selected life insurance markets have great commonalities, to understand their respective macroeconomic environment and micro market environment more accurately, we have compiled statistics and per-formed simple calculations on the relevant data. Some of the data selected in this section are also used as environmental variables in the three-stage DEA model. 2.1 Macroeconomic Environment In this paper, we select the population and gross domestic product (GDP) growth rates as indicators to measure the macroeconomic environment. Generally speaking, there is a positive relationship between the growth rate of the GDP and the efficiency of the life insurance market, but it is difficult to judge the impact of the population directly. We select the complete data from 2013 to 2019 and present it in the form of a figure. • Population The main function of insurance is to transfer risk. Different from other types of insurance, life insurance transfers the risk of survival or death of the insured. At first, life insurance was only used to protect the economic burden caused by unpredictable death. Later, life insurance introduced the element of saving, and gradually it became an investment tool with both insurance and saving functions. The population is an important index that is closely related to the life insurance market. If we want to study the relationship between the population and the life insurance market carefully, we need to analyse it according to age, gender, urban/rural location, education level, and other aspects. The current focus of this paper is not on these aspects, so, in this section, we only compare the total population of different markets. The results are shown in Figure 2-1. Figure 2-1 The Population of the Selected Life Insurance Markets According to the population data, from 2013 to 2019, no matter which insurance market is targeted, the population has no fluctuation and maintains a stable level. Thus, we only show the results for 2019. Generally speaking, countries with a less serious aging 0 10 20 30 40 50 60 70 80 90 million people Population-2019 58 Ekonomická revue – Central European Review of Economic Issues 24, 2021 problem will have a more stable population. After the outbreak of COVID-19 in 2020, the population may change considerably, but we do not have the relevant data yet. Germany has the largest population, fol-lowed by Italy, Spain, and Poland. The population of Luxembourg is very small compared with that of the other countries, but the density of its life insurance market is very large, and we will introduce it in the next section. • GDP Growth (%) The GDP growth rate is one of the four important macroeconomic indicators (the other three are the unemployment rate, inflation rate, and balance of payments). It usually reflects the economic growth level of a country or region. The development of the insurance industry and economic growth are closely related and interact with each other (Enz, 2000). On the whole, the insurance industry will develop better and faster in countries with fast economic develop-ment; the insurance market activities also have a causal relationship with economic growth. Arena (2008) found that both life insurance and non-life insurance have a positive and significant causal effect on economic growth. The results are shown in Figure 2-2. Figure 2-2 GDP Growth Rate of the Selected Life Insurance Markets From Figure 2-2, we can see that, different from the population, the GDP growth rate fluctuates greatly from 2013 to 2019. Belgium, Denmark, Germany, Hungary, Luxembourg, and Poland all have positive GDP growth during this period, and the fluctuation is relatively small. However, it is worth noting that Ireland’s GDP growth rate is also positive, but it has a very significant surge in 2015, as high as 25%. In recent years, the development of Ireland’s emerging industries has been rapid, and it has been given the title of the “European Silicon Valley.” It has kept pace with the development of the information and com-munications technology (ICT) industry, the rapid development of the information industry helping to increase its GDP. Finland, Greece, Italy, Portugal, and Spain sometimes show negative GDP growth in the past seven years. Among them, Greece has the worst economic growth, experiencing three times negative growth, and the lowest GDP growth rate is about -3%. Generally speaking, in recent years, the GDP growth rate of these countries follows a downward trend, and only Denmark and Belgium maintain their growth at the previous level. 2.2 Life Insurance Industry Environment Regarding the environment of the life insurance market, we choose four indicators, the total gross premiums, density, market share in the OECD, and penetration. Different from the previous section, in this section, we calculate the arithmetic mean of relevant data for each life insurance market from 2013 to 2019 for analysis and comparison. To com-pare different markets more intuitively and clearly, we do not use panel data in this section. • Total Gross Premiums To a certain extent, the total gross premium can reflect the scale of the whole insurance market. Usually, a higher total gross premium means a greater market share and a larger market scale. Figure 2-3 shows the total gross premiums. Figure 2-3 Total Gross Premiums of the Selected Life Insurance Markets -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 Belgium Denmark Finland Germany Greece Hungary Ireland Italy Luxembourg Poland Portugal Spain GDP Growth (%) 2019 2018 2017 2016 2015 2014 2013 B. Guan – Efficiency comparison of selected OECD life insurance markets using a three-stage DEA model 59 From Figure 2-3, it is not difficult to see that the Italian life insurance market has the largest total gross premium, while the German life insurance market ranks second with a small gap, which is somewhat surprising. Germany’s insurance market is supposed to be the largest in terms of scale, far larger than other insurance markets. It is apparent that the German life insurance market no longer has obvious advantages this year. In addition, we can see that the total gross premiums of Greece and Hungary are the lowest. What is interesting here is that, although Luxembourg has the smallest population, its total gross premiums are in the middle of the 12 life insurance markets. Luxembourg’s per capita wage is very high, the unemployment rate is very low, and its medical insurance system is perfect, the average life expectan-cy being as high as 83 years. These could be the reasons for its high total gross premium. • Density The density is obtained by dividing the direct gross premium by the population, which can reflect the development degree of the insurance business and the strength of people’s insurance consciousness. The results are shown in Figure 2-4. Figure 2-4 Density of the Selected Life Insurance Markets Although the total gross premiums of each market are quite different, when we compare the density of the markets, we find that it is very similar, except in Denmark, Ireland, and Luxembourg. In particular, the density of the life insurance market in Luxembourg is much higher than that of other life insurance markets. To a certain extent, this shows that the insurance industry in Luxembourg has a high degree of devel-opment and people have a strong sense of insurance. As we mentioned earlier, Luxembourg’s medical insurance system is perfect, and the statutory health system is responsible for 99% of residents’ health care. • Market Share in the OECD Generally, the larger the market share, the stronger the competitiveness. However, a large market share does not mean a high profit as it is only one of the factors affecting profitability. There are several ways to calculate the market share; here, we use the direct gross premium basis. The results are shown in Figure 2-5. Figure 2-5 Market Share of the Selected Life Insurance Markets The market share of the Italian life insurance market is the largest. As we mentioned earlier, there is a positive relationship between the total gross premi-um and the market share, which is confirmed here. The Greek life insurance market and the Hungarian life insurance market are ranked second to last and last, respectively. If we compare Figure 2-5 with Figure 2-3, we find that the rankings of each market are almost the same. • Penetration The penetration is obtained by dividing the direct gross premium by the GDP, which reflects the position of the insurance industry in the whole national economy. Penetration usually depends on a country’s overall economic development level and the devel-opment speed of the insurance industry. Penetration can also be used to judge the development potential of the insurance market. The results are shown in Figure 2-6. 0 5 10 15 20 25 30 35 40 Density (thousands of US Dollar) 0,745 2,076 0,256 4,348 0,087 0,070 1,537 4,947 0,908 0,2880,381 1,316 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 (%) 60 Ekonomická revue – Central European Review of Economic Issues 24, 2021 Figure 2-6 Penetration of the Selected Life Insurance Markets From Figure 2-6, we can see that the penetration of the life insurance market in Luxembourg is the highest, close to 35%. The second is the penetration in the life insurance market in Ireland, which is about 12.5%. Germany and Italy have the top two life insurance markets in terms of the total premium and market share, but their penetration is not high, with 2.86% and 6.057%, respectively. The last two are still the Hungarian life insurance market and the Greek life insurance market. When we compared the GDP growth rate earlier, we found that Greece’s economic growth is the worst, which is one of the reasons for the poor development of its life insurance market. 3. Data and Methodology In the second stage of the DEA model, in addition to the input and output variables required in the traditional model, we need to select environmental variables. These variables must not be controlled personally but also need to affect the efficiency score. In this paper, we consider the insurance market environment and the macroeconomic environment. Thus, we choose the insurance density, market share, and growth of the GPD as the relevant indicators. Huang and Eling (2013) also selected the ratio of shareholder equity to assets, liabilities to liquid assets ratio, and premiums to surplus ratio as indicators of the insur-ance market’s regulatory environment. In the existing research, the choice of input variables and output variables has also been very different. Kaffash et al. (2019) found that, as output variables, “premiums” accounted for 50.82%, “losses and incurred losses” accounted for 22.13%, and “investment income” accounted for 21.31%, while, as input variables, “the number of employees”, “capital debt”, “equity capital”, and “materials and business ser-vices” accounted for 60.72%, 49.18%, 37.7%, and 32.79%, respectively. The variables that we selected are slightly different from the above variables, which we will explain in detail in the next section. 3.1 Data This paper selects the relevant data of 12 OECD life insurance markets from 2013 to 2019, mainly from the OECD (2014–2020). When using the three-stage DEA model, it is very important to define the input, output, and environmental variables reasonably. Different from other manufacturing industries, the products of the insurance industry are invisible, which makes it more difficult to define and evaluate input and output variables than in other industries. The rationality of the environmental variables will also affect the significance of the existence of the second stage. • Input Variables The input variables that have generally been used in the previous studies can mainly be divided into three categories, namely labour input, capital input, and other material input (Eling and Jia, 2019; Eling and Luhnen, 2010; Eling and Schaper, 2017). How-ever, we could not find the number of employees in only the life insurance market; thus, in this paper, the number of companies, debt capital, and equity capital are chosen as the input indicators. Many previous studies have used the price of labour to represent the labour input (Cummins et al., 2004; Huang and Eling, 2013), choosing the employee wages in the financial and insurance services as one of the input variables. In this paper, we do not apply this approach because these employee wages do not accurately represent the labour price of the life insurance industry. The number of companies can represent the competitive ability of the insurance industry; the other two indicators can represent the capital input and operational situation of the industry. • Output Variables Regarding the output variables, we can consider the social functions of the insurance industry. One of the very important functions undertaken by insurance is risk protection. The non-insurance industry is more concerned about the claims, while the life insurance industry is more concerned about the benefits. Therefore, in this paper, for the life insurance industry, we choose the sum of the net income plus gross technical provisions and the total investments as the output variables. • Environmental Variables 3,626 7,473 2,4452,860 1,015 1,238 12,505 6,057 34,503 1,360 4,263 2,441 0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00 (%) B. Guan – Efficiency comparison of selected OECD life insurance markets using a three-stage DEA model 61 In the selection of environmental variables, we consider the macroeconomic environment of the whole market and the industry environment of the life insurance market itself. We choose the growth of the GDP (related to the overall economy), the insurance density (related to the insurance market), and the market share (related to the insurance market) as related variables. The growth of the GDP can show us the quality of the macroeconomic environment in which the market is located. Generally, rapid GDP growth is more conducive to an efficient state of the insurance industry. Through the insurance density and market share, we can understand the importance and development of each individual insurance market in the whole industry. According to our previous re-search, the effective value of an insurance market with a larger market share or higher insurance density is usually higher. These environmental variables will eventually have a greater impact on the efficiency score. Table 3-1 presents the sample summary statis-tics. Table 3-1 Summary of the Sample Statistics of the 12 Life Insurance Markets Unit Min Mean Max Std. dev. Panel A: Input variables Number of companies 1 2 35.10 93 24.61 Debt capital Million US dollars 2761 215989 126429 0 329010 Equity capital Million US dollars 167 6927.70 25254 7321.54 Panel B: Output variables Total investments Million US dollars 1407 197433 142791 3 341112 Net income + Technical provisions Million US dollars 2225 194739 120690 3 304032 Panel C: Environmental variables Density US dollars 141 4953.74 48768 10536 Market share % 0.10 1.41 8.40 1.74 Growth of GDP % - 3.241 2.556 25.163 3.203 Number of observations 672 From Table 3-1, we can see that, of the input variables, the degree of dispersion of the debt capital is very large; the degree of dispersion of both the output variables is very large; and, in the environment varia-bles, the degree of dispersion of insurance density is the largest. Through the observation of the results in Table 31, it is not difficult to find that the maximum values of most of the indicators are far higher than their average values. Looking at the original data, we can see that this is because the input and output values of the German life insurance market are much higher than those of the other markets, which is also the main reason for the wide dispersion of various indicators. At the same time, we find that equity capital is much lower than debt capital, while the two output variables are close. 3.2 Three-Stage DEA Model Data envelopment analysis is suitable for the evaluation of complex multi-output and multi-input problems. We can use the DEA model to calculate many kinds of efficiency scores. In this paper, we mainly focus on the TE, PTE, and SE of the life insurance industry. Table 3-2 describes these three kinds of efficiency in detail. Table 3-2 DEA Efficiency Terms Term Description Decomposition Technical Efficiency TE reflects the ability of a manufacturer to maximize output under a given input, the returns to scale are fixed. ( θ from CCR model) TE=SE×PTE Pure Technical Efficiency PTE reflects the production efficiency of the inputs of the DMU at the optimal scale, the returns to scale can be changed. ( θ from BCC model) PTE=TE/SE Scale Efficiency SE reflects the gap between the actual scale and the optimal production scale. SE=TE/PTE • The First Stage: Calculate Efficiency using Unadjusted Input or Output Variables In this paper, we select the input-oriented BCC model (Banker et al., 1984) and the input-oriented CCR model (Charnes et al., 1978) to calculate the required efficiency. The assumption of the CCR model is that, in the production process, the scale return is fixed. When the input changes in proportion, the output should also change in proportion. For the input-oriented CCR model, the optimization model is as follows: min θ (1) s.t. ∑ λjxij ≤ θxi0 n j=1 (2) ∑λjyrj ≥ yr0 n j=1 (3) λj ≥ 0, i = 1, 2 , ..., n; j = 1, 2, ..., n; r = 1, 2, ..., n (4) where xij represents the i-th inputs of the j-th DMU and yrj represents the r-th outputs of the j-th DMU; here, there are three inputs, two outputs, and 12 DMUs. λj is a scalar, and θ is an input radial measure of technical efficiency. Among them, the optimal solution is θ *, and 1-θ * represents the maximum input that can be reduced without reducing the output level at the current technical level. A larger θ * means that a smaller amount of input can be reduced, representing greater efficiency. When θ * = 1, it means that the DMU is currently in a technical effective state. The BCC model has almost the same constraints as the CCR model. The only difference is that, in the BCC model, there is also a constraint on λ, which can 62 Ekonomická revue – Central European Review of Economic Issues 24, 2021 basically ensure that manufacturers of a similar size are compared with manufacturers that are not valid rather than manufacturers with large gaps. The constraint is as follows: ∑𝜆𝑗=1 𝑛 𝑗=1 (5) • Second Stage: Adjusting the Input or Output Varia-bles with SFA Slack Regression When using SFA slack regression to regress the slack variables in the first stage, we need to consider whether to adjust the input and output variables at the same time or to adjust only one of them. Fried et al. (2002) proposed that this depends on the type of orientation that we choose in the first stage. In this paper, we choose the input-oriented approach, so, in the second stage, we only adjust the input variables. In addition, Fried et al. (2002) mentioned that we should perform a separate regression for each differ-ent slack variable, which allows the environmental variables to have different effects on different slack variables. We can construct the following SFA slack regression functions: Sni = f (Zi; βn) + vni + μni (6) i = 1, 2, ... , I; n = 1, 2, ... , N (7) where Sni is the slack value of n-th inputs on i-th DMU; Zi represents the environmental variables, βn represents the coefficient of environmental variables; vni represents the statistical noise and μni represents the managerial inefficiency. v ~ N (0, σv2) is the random error term, it can represent the influence of statistical noise on input slack variables; μ ~ N+ (0, σμ2) can represents the influence of managerial inefficiency on input slack variables. As mentioned earlier, using SFA slack regression helps to eliminate the influence of statistical noise and environmental effects. Therefore, we need to adjust the input variables. The adjustment formula is as follows: Xni A = Xni + [ max( f (Zi; β n))- f (Zi; β n) ] + [ max(vni) - vni ] (8) i = 1, 2, ... , I; n = 1, 2, ... , N (9) where Xni A represents the adjusted input variables; Xni is the original input variables; [ max (f (Zi; β n)) - f (Zi; β n) ] represents the adjustment of the input variables based on the environmental effects; f is the function form; [ max (vni) - vni ] shows the adjustment of the input variables based on the statistical noise. To calculate the statistical noise, we have the following formulas: 𝐸(𝜇|𝜀 ) = 𝜎 ∗ × [ 𝜙(𝜆𝜀 𝜎) 𝛷 (𝜆𝜀 𝜎) + 𝜆𝜀 𝜎 ] (10) σ*=σμσv σ (11) σ= √ σμ2+σv2 (12) λ= σμ σv (13) 𝐸 [ 𝑣𝑛𝑖|𝑣𝑛𝑖 + 𝜇𝑛𝑖 ] = 𝑆𝑛𝑖 − 𝑓 (𝑍𝑖; 𝛽𝑛) − 𝐸 [ 𝜇𝑛𝑖|𝑣𝑛𝑖 + 𝜇𝑛𝑖 ](14) We predict the maximum slack to set up a base equal to the worst external conditions. When the predicted slack is lower than the maximum predicted slack (among all the DMUs), we increase the inputs; if the DMU has very good conditions, after the adjust-ment, it may lower their efficiency. • Third Stage: Calculate the Efficiency Using Adjusted Input or Output Variables In this stage, the adjusted input variables are re-applied to the DEA model of the first stage to obtain a new efficiency score. The adjusted results will be more accurate than the results of the first stage because all the DMUs are adjusted to the same external environ-ment, and the impact of statistical noise is proposed. 4. Empirical Results In the first stage, using DEAP 2.1 can help us to obtain the initial efficiency score of the TE, PTE, and SE; then, in the second stage, we can adjust the input variables with Frontier 4.1, using the input orientation and selecting the cost function; in the third stage, we use the adjusted input variables and employ DEAP 2.1 to recalculate the adjusted efficiency score. There are 12 life insurance industries as DMUs, and the period is seven years, from 2013 to 2019. When we use DEAP 2.1 to calculate the efficiency score, we first need to choose the specific model to use. We select 12 decision-making units for seven years. Unlike the direct calculation of section data, when we use panel data, we have two options. We can split the panel data into cross-sectional data and calculate the efficiency scores, respectively, and summarize them or we can use the Malmquist model, entering the panel data directly. The Malmquist model (Fare et al., 1992) can be used to measure the productivity change, and the productivity change can be decomposed into tech-nical change and technical efficiency change. Through the Malmquist model, we can also obtain the TE and PTE of each market in every year, and then we can calculate the SE. However, there is a problem: when we use the three-stage DEA model, in the second stage, to adjust the input variables, we need to use the input slacks. The Malmquist model cannot produce input slacks. Thus, we choose the first method of dealing with the application of panel data in DEAP 2.1. 4.1 Stage 1 In the first stage, we choose a multi-stage DEA model with input-oriented and variable scale returns. The B. Guan – Efficiency comparison of selected OECD life insurance markets using a three-stage DEA model 63 multi-stage model is more accurate than the other models. We calculate the technical efficiency, pure technical efficiency, and scale efficiency of each market in each year and calculate the arithmetic mean value of the relevant efficiency of each market from 2013 to 2019; we summarize all the results in Table 4-1. At the same time, Figure 4-1 shows the average efficien-cy of each market to provide a clearer comparison of the efficiency of the different markets. Table 4-1 Efficiency Score from Stage 1 TE PTE SE TE PTE SE Belgium Denmark 2013 0.933 1 0.933 0.942 0.945 0.997 2014 0.919 0.965 0.952 0.92 0.961 0.957 2015 0.904 1 0.904 0.916 1 0.916 2016 1 1 1 0.944 0.979 0.964 2017 1 1 1 0.966 0.966 1 2018 1 1 1 0.984 0.991 0.993 2019 0.973 0.981 0.992 0.993 1 0.993 Finland Germany 2013 1 1 1 1 1 1 2014 1 1 1 1 1 1 2015 1 1 1 1 1 1 2016 0.983 0.996 0.987 1 1 1 2017 0.967 0.975 0.992 1 1 1 2018 1 1 1 1 1 1 2019 0.998 1 0.998 1 1 1 Greece Hungary 2013 0.965 0.985 0.98 0.951 1 0.951 2014 0.44 0.927 0.475 0.947 1 0.947 2015 0.448 0.925 0.485 0.952 1 0.952 2016 0.989 1 0.989 1 1 1 2017 1 1 1 1 1 1 2018 0.989 1 0.989 0.999 1 0.999 2019 0.992 1 0.992 1 1 1 Ireland Italy 2013 1 1 1 0.957 0.975 0.982 2014 1 1 1 0.958 0.973 0.984 2015 1 1 1 0.961 0.979 0.982 2016 0.995 0.995 1 1 1 1 2017 0.995 1 0.995 1 1 1 2018 0.987 0.995 0.992 0.919 0.919 1 2019 0.985 0.992 0.993 1 1 1 Luxembourg Poland 2013 0.958 0.963 0.994 0.958 0.962 0.996 2014 0.963 0.97 0.994 0.953 0.957 0.995 2015 0.954 0.962 0.992 0.967 0.972 0.995 2016 0.999 1 0.999 1 1 1 2017 1 1 1 1 1 1 2018 1 1 1 1 1 1 2019 1 1 1 1 1 1 Portugal Spain 2013 0.974 0.977 0.997 0.937 0.989 0.948 2014 0.969 0.972 0.997 0.887 0.941 0.942 2015 0.967 0.97 0.997 0.925 0.982 0.942 2016 0.985 0.985 1 0.938 0.939 0.999 2017 0.985 0.986 1 0.926 0.943 0.981 2018 0.926 0.926 0.999 0.917 0.917 1 2019 0.873 0.877 0.995 1 1 1 From Table 4-1, we can gain a very detailed un-derstanding of the relevant efficiency score of each life insurance market in each year. We observe that almost all the markets reach the effective state between 2013 and 2019, with the exception of Den-mark and Portugal. In addition, we can see that there are more efficient markets in 2015–2018 than in 2013–2015. As many as seven life insurance markets, more than half of them, had reached an effective state in 2017. Figure 4-1 Comparison of the Results from Stage 1 Although, in the previous chapter, we found that the German life insurance market is not the best in all aspects, its whole market reached an effective state between 2013 and 2019, which shows that, in the German life insurance markets, input resources are not wasted and all inputs are completely and effec-tively converted into output. Of the remaining 11 life insurance markets, Greece’s efficiency is the lowest, with technical efficiency of only 0.832 and scale efficiency of only 0.844. However, the scale efficien-cy of Greece is not the lowest; Poland has the lowest SE score of only 0.956. The pure technical efficiency scores of the markets are not very different, and one of the reasons is that, when calculating the PTE, the returns to scale are variable. In addition, the TE value obtained in the first stage is 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 TE PTE SE Belgium Denmark Finland Germany Greece Hungary Ireland Italy Luxembourg Poland Portugal Spain