Efficiency and Scale Properties of Indian General Insurers

Abstract

The current research applies a multi-stage approach for measuring and explaining the efficiency performance of 20 general insurers operating in India for the phase 2012–2013 to 2019–2020. In the first stage, the study adopts non-parametric radial data envelopment analysis (DEA) for point and interval estimation of firm-specific efficiency, scale efficiency, returns to scale (RTS) and scale elasticity. The second stage of the study applies panel data regression for regressing technical and scale efficiency scores on the index of market concentration, insurer age, return on shareholders’ capital and the solvency indicator. The outcome of the next stage indicates that the index of market concentration and insurer age are the two contextual variables which are statistically significant, although their impacts on technical and scale efficiency are negative. The influence of the solvency ratio is significant for scale efficiency only.

JEL Classifications: C-23, C-61, D-22, G-22

Keywords

General insurance data envelopment analysis panel data regression

Introduction

In the domain of neoclassical production economics, the relationship between the inputs and output of the production process is often expressed in terms of a functional relationship between the quantity of output produced by the firm and the quantities of inputs used by it. The neoclassical production function is assumed to be well behaved, exhibiting homotheticity, smooth substitutability of inputs and diminishing marginal productivity. A producer can produce sub-optimally in the short run because capital is not a variable factor (in the short run). A homothetic production function is a monotone transformation of a homogeneous function. Shephard introduced the concept of duality, which implies that the production and cost functions are dual representations of the underlying production technology so that one of them can be obtained from the other one (and vice versa). The concept of duality in production economics is actually based on Fenchel (1949), which showed that the conjugate of a convex function is also convex.

An alternative approach to the theory of production function was introduced by Frisch in the form of an s-shaped technology which is more general in nature, as it accommodates three phases of returns to production scale (RTS) as the technology displays IRS (increasing returns to scale) in the initial stage of production followed by phases of CRS and DRS (constant and decreasing returns). Thus, compared to the neoclassical production function, which shows a Passus coefficient (elasticity of output relative to changes in the returns to scale [RTS]) varying between 0 and 1, the s-shaped production function can have a segment for which the Passus coefficient exceeds unity.

Given the aforementioned theoretical background, the current study seeks to estimate and analyze the technical efficiency and scale efficiency of in-sample non-life (general) insurance providers for a time span covering 2013–2014 to 2019–2020. In the post-reform phase, none of the existing studies provides an explicit link between the neoclassical microeconomic framework and the empirical evaluation of efficiency and scale properties. The current study seeks to fill this gap. In specific, the study seeks to address the below-mentioned (research) questions:

RQ1: Exploration of the trends in efficiency, RTS and (scale) elasticity of the Indian general insurers during 2012–2013 to 2019–2020.

RQ2: Impact of the exogenous (contextual) variables on the (technical and scale) efficiency estimates for the aforementioned period.

The current study can be divided into two stages. In the first stage, the study seeks to estimate technical and scale efficiency, RTS and the degree of scale economies of the observed general insurers. In the second stage, we have explored the relationship of estimated efficiency with the contextual variables selected for the study. The remaining part of the study has five sections and progresses as follows. Next, section ‘An Inspection of the Prior Studies’ reviews the extant literature on insurer efficiency. ‘Methodology’ section discusses the conceptual framework required for efficiency evaluation. This section also discusses the envelopment methodology of data (data envelopment analysis [DEA]) utilized in this paper for the estimation of efficiency (technical and scale), RTS and scale elasticity. ‘The Evaluation Framework’ section describes the data and estimation outcomes. ‘The Concluding Remarks’ section includes the concluding remarks.

An Inspection of the Prior Studies

International Studies

Extant non-life insurance performance literature in the international context encompasses areas like the presence of economies or diseconomies of scale, the impact of market and firm-level unification and the impact of product diversification. While the majority of the studies considered a one-stage black box model of production, some of the research studies are based on a network structure of production.

International research studies undertaken in the nineties and the early phase of the current millennium estimated productivity, efficiency and RTS mostly from the cost or profit side. These include Weiss (1991) which estimated a Leontief type profit function of property liability insurers for 1980–1984, Cummins and Weiss (1993) stochastic cost frontier for property-liability insurance companies for the period 1980–1988, Rai (1996) which estimated cost efficiency of insurance companies across 11 countries for 1988–1992, and Toivanen (1997) which estimated the cost efficiency frontier of Finnish non-life insurers for the phase 1989 through 1991 to investigating the presence of economies of scale and scope. Fukuyama and Weber (2001) estimated efficiency and productivity growth of 17 Japanese non-life insurers for the 11-year period 1983–1994. Ferro and León (2017) estimated comparative technical efficiency and technical change of Argentinian general insurers for the phase 2009–2014 using the parametric (stochastic) frontier approach.

A second group of studies examined the impact of various contextual variables like deregulation, market unification or market structure. Ennsfellner et al. (2004) examined the impact of deregulation (1994–1999) on the Austrian insurance sector and found a positive relationship between market deregulation and insurer efficiency. Choi and Weiss (2005) applied the stochastic frontier approach for examining the relationship between performance (of the firm), structure of the market and (cost and revenue) efficiency among US general insurers during 1992–1998. Barros et al. (2010) evaluated the impact of competition on Greek insurance companies (non-life and life) for 1994–2003 using a conditional DEA model. Mahlberg and Url (2010) considered the influence of market integration on 202 German insurance companies for 1991–2006. Cummins and Xie (2013) examined the influence of insurer size on efficiency and productivity. Alhassan and Biekpe (2015) evaluated efficiency, total factor productivity and scale properties of South African general insurers for 2007–2012 and found a nonlinear impact of insurer size on productivity and efficiency. Kramarić et al. (2022) investigated the efficiency drivers of Croatian non-life insurers for 2009–2021 and found that ownership and insurer age significantly influenced insurer performance.

Yang (2006) utilized a multi-stage non-parametric model for evaluating both production and investment efficiency of the health and life insurers of Canada. Kao and Hwang (2008) used multi-stage DEA models for 24 non-life insurers of Taiwan (based on mean observations for 2001 and 2002).

Indian Studies

Recent efficiency studies related to the Indian general insurers include Sinha (2007), Mandal and Dastidar (2014), Sinha (2017a, 2017b), Ilyas and Rajasekharan (2019a, 2019b) and Sinha and Vaisi (2022). Mandal and Dastidar (2014) evaluated the efficiency of several insurers in India for the span 2006–2007 through 2009–2010 utilizing DEA. The study evaluated the impact of the worldwide economic recession on the Indian insurers. Sinha (2017a) applied a dynamic DEA approach for evaluating intertemporal efficiency for the Indian non-life (general) insurers, while Sinha (2017b) linked performance with insurer solvency. Two studies of Ilyas and Rajasekharan (2019a, 2019b) estimated technical and scale efficiency and change in productivity of insurers for 2005–2016. The first study by Ilyas and Rajasekharan (2019a) estimated technical, scale, cost and allocative efficiency. The second study by Ilyas and Rajasekharan (2019b) estimated the Fare–Primont index of productivity and found total factor productivity in the sector. The total factor productivity growth observed during the observed period (2005–2016) is mainly due to scale and mix inefficiency.

Motivating Factors Behind the Current Research

While revisiting the issue of efficiency and scale properties of Indian general insurers, this study makes several (value) additions to the prior literature. First, as mentioned in the outset, it explicitly discusses the theoretical framework which serves as the backbone of the process of efficiency and scale evaluation. For facilitating intertemporal comparison of performance, the current study utilized panel data, leading to the construction of a global frontier of performance. Finally, in addition to the assessment of RTS and scale efficiency, the present study estimates the degree of economies of scale of the general insurers so that the responsiveness of output in response to changes in the RTS can be analyzed.

Methodology

In our study, the radial one-stage DEA approach is adopted over other methods for making an elaborate evaluation of the scale properties of the insurers. The current section briefly indicates the details of the estimation process of the production frontier and firm-wise efficiency under the radial approach.

Technical Efficiency

In the context of a production technology, the concept of technical inefficiency was introduced with the aid of the idea of distance functions introduced by Shephard (1953, 1970, 1974) and Farrell (1957). The (distance) functions permit a comparison of observed points with the best practice (efficient) points lying on the frontier. Corresponding to the input and output sides, we can represent the two distance functions as:

D_{Shephard}^{Input} = Max \{θ : (\frac{X}{θ}, Y) \in T\}

D_{Shephard}^{Output} = Min \{ϕ : (X, \frac{Y}{ϕ}) \in T\}

The input-oriented and output-oriented distance functions provide the maximum (minimum) amount by which the observed input (output) quantity can be divided and yet the target output (input) is attainable. Farrell’s distance functions (1957) can be obtained by inverting the distance function of Shephard (1953). In specific, both Shephard (1953) and Farrell (1957) introduced efficiency measures based on a proportional reduction of input/increase of output compared to the production frontier. Further, Farrell identified the frontier as the (maximum) pessimistic linearized envelopment of observed data points in the context of a one-output CRS technology.

Charnes et al. (1978) extended the approach advocated by Farrell in the context of several outputs and inputs but retained the assumption of CRS. Banker (1984) further extended the programming linked data envelopment approach in the context of varying returns.

Scale Efficiency

Relating to a production relation Y = f(X) where X and Y denote the input and output vectors, suppose the inputs are multiplied by α (the revised input vector is αX and the resultant output vector is βY). Banker (1984) showed that the return to scale can be measured at a point on the radially efficient surface of the production (possibility) set as 𝛿 = $\lim_{β \to 1} \frac{α - 1}{β - 1}$ . Here 𝛿 can exceed, equal or fall short of 1 depending on the existence of increasing, constant or decreasing RTS.

Formal Presentation of the Evaluation Methodology

For explaining the Banker et al. (2019) model of evaluation, we invoke a hypothetical technology featuring several outputs and inputs. The vectors of input and output are represented by $X = (x_{1}, x_{2}, \dots, x_{m})$ and $Y = (y_{1}, y_{2}, \dots, y_{s})$ , respectively. There are n firms’ decision-making units (DMU) to be evaluated. Each DMU _j uses $X_{i} = (x_{1 i}, x_{2 i}, \dots, x_{m i})$ to produce $Y_{j} = (y_{1 j}, y_{2 j}, \dots, y_{s j})$ . The production possibility/technology set T, which includes feasible input-output transformation plans of the DMUs, can be represented as:

T = \{(X, Y) \in R^{m + s}, X can produce Y\}

The technology set includes output as well as input sets. The output set $[O (Y) \in T]$ includes the feasible outputs (for given input vectors), while the input set $[I (X) \in T]$ includes the vector of feasible inputs for given output vectors.

The following axioms are satisfied by the technology:

The technology set T is closed and non-empty, that is, it includes all of the boundary points and is non-empty.

Strong disposability of inputs and outputs: If $(X, Y) \in T$ , then free disposability implies that $(X^{'}, Y) \in T$ , where $X^{'} \geq X$ . Similarly, if $(X, Y) \in T$ , then $(X, Y^{'}) \in P$ , where $Y \geq Y^{'}$ .

Convexity: The convexity of the technology set implies that if $(X, Y) \in T$ . and $(X^{'}, Y^{'}) \in T$ then $(X^{c}, Y^{c}) \in T$ , where $X^{c}$ is a weighted combination of (X,Y) and $(X^{'}, Y^{'})$ . Thus $X^{c} = ω X + (1 - ω) X^{'}$ . Similarly for Y. Here 0 < 𝝎 < 1.

Minimum extrapolation of data: A minimal set is not a proper subset of any other set. This implies that the technology set (T) is constituted by the minimum extrapolation of data.

Taking the four axioms into consideration, the technical form of the empirical production possibility set T is provided here:

T = \{(X, Y) : \sum_{j = 1}^{n} λ_{j} x_{j} \leq x_{0}, \sum_{i = 1}^{n} λ_{j} y_{j} \geq y_{0}, \sum λ_{j} = 1, λ_{j} \geq 0, j = 1, 2, \dots, n\}

Thus, the input and output-oriented measures of technical efficiency are:

E_{input} (X, Y) = E_{input} X \geq \sum_{j = 1}^{n} λ_{j} x_{j}, \sum λ_{j} = 1

E_{output} (X, Y) = E_{output} Y \leq \sum_{j = 1}^{n} λ_{j} y_{j}, \sum λ_{j} = 1

In the Charnes et al. approach of 1978, the assumption of global (constant) RTS rules out scale variations. The addition of the convexity constraint and minimum extrapolation of data to the Charnes et al. model allows local variations in technology, and thus the (pure-technical) efficiency measure is essentially a local measure of efficiency. The global measure of efficiency Charnes–Cooper–Rhodes (CCR), on the other hand, occurs at a point where average productivity is maximum most productive scale size (MPSS). The locally efficient point is also globally efficient if the firm is scale efficient, that is, it attains MPSS.

Banker (1984) derived the most productive scale size for a particular input and output mix and estimated the RTS. Banker and Thrall (1992) extended the RTS measure in the case of an MPSS interval accommodating the possibility of multiple optimal solutions. The frontier can thus be partitioned into segments exhibiting, increasing, constant and decreasing RTS.

In the radial approach to efficiency evaluation, scale efficiency is estimated from the CCR and Banker–Charnes–Cooper (BCC) measures of efficiency. Scale efficiency is derived as the ratio of the CCR and BCC measures of efficiency. Observed firms are scale efficient when both measures are equal, implying that scale efficiency is unity. In case of non-constant (increasing or decreasing) RTS, the firm has scale efficiency scores that are lower than unity.

Estimation of Scale Elasticity

Färe et al. (1988) proved that scale elasticity can be obtained from four approaches, all of which provide equivalent measures of scale elasticity.

The first measure utilizes the transformation function [T(Y,X)] and was introduced by Panzar and Willig (1977):

E (y, x) = \frac{\sum_{j = 1}^{m} x_{i} \frac{\partial T}{{\partial x}_{i}}}{\sum_{i = 1}^{s} y_{i} \frac{\partial T}{{\partial y}_{i}}}

Another measure of scale elasticity can be obtained from the output distance function.

E (y, x) = \frac{1}{D_{o u t p u t} (y, x)} \sum_{i = 1}^{m} x_{i} \frac{{\partial D}_{o u t p u t} (y, x)}{\partial x_{i}}

Fare et al. (1988) provided a third measure of scale elasticity based on the output distance function.

{E (y, x) = - [\frac{1}{D_{I n p u t} (y, x)} \sum_{i = 1}^{s} y_{i} \frac{{\partial D}_{I n p u t} (y, x)}{\partial y_{i}}]}^{- 1}

The fourth measure of scale elasticity is derived from the cost function.

{E (y, p) = [\frac{1}{C (y, p)} \sum_{i = 1}^{s} y_{i} \frac{\partial C (y, p)}{{\partial y}_{i}}]}^{- 1}

Here, the cost function is denoted by C(y,p), where p is the input price vector. Thus, the cost function-based measure of scale elasticity measures the responsiveness of firm output to variations in the cost of production.

Relationship of Efficiency with Explanatory Variables

The (efficiency) performance of the observed insurers is dependent on several contextual factors which (indirectly) influence the process. In the current case, four contextual variables (index of market concentration, insurer age, solvency ratio and return on equity) are used as explanatory variables. Insurer age is a proxy for insurer experience. Solvency ratio is a measure of capital adequacy. Return on equity indicates profit earned per unit of capital and is expected to have a positive relationship with efficiency.

For estimating the relationship of efficiency with the exogenous variables, a panel data approach has been used. The benefit from the panel data framework arises from its ability to capture the unobserved effects of time and firm-specific factors. We have used logarithmic transformation of efficiency estimates as dependent variables (based on Banker and Natarajan (2008), as well as Banker et al. (2019)).

The Evaluation Framework

Choice of Input, Output and Exogenous Variables

Frontier efficiency evaluation needs the selection of inputs, outputs (and prices) for revenue/cost/profit efficiency models. Eling and Luhnen (2010) found three major types of inputs used in the insurance industry: agents and employees, business development and financial capital (including borrowed and owned capital). Regarding the output choice, Leverty and Grace (2010) found three methods: the (financial) intermediation approach (Sealey & Lindley, 1977), the user cost method (Hancock, 1985, 1991) and the value-addition approach (Berger et al., 1987; Berger & Humphrey, 1992). The intermediation approach treats financial service providers as intermediaries. The user-cost method by Hancock (1985, 1991) considers an indicator as an input or an output depending on whether its net revenue contribution is positive or negative. The third (value-added) approach considers such activities as outputs which lead to value addition (Berger et al., 1987). The value-added approach treats the insurers as the providers of the following services: risk pooling, bearing, real financial services and intermediation.

The present study is based on a revenue frontier, which needs data on inputs, outputs and the corresponding prices. Here, we have included two composite inputs: total operating expenses (a proxy input for capturing operations) and net worth (an indicator of capital contributions of equity shareholders). Moreover, we have considered premium income (net of commissions) and investment-related income as the output indicators. All the indicators are in value terms. Thus, revenue efficiency is equivalent to technical efficiency. In the subsequent analysis, four environmental (context related) variables (market share of the general insurers, the solvency indicator ratio, return on shareholders’ capital and insurer age) have been considered. Table 1 presents a brief statement of the input indicators, output indicators and explanatory variables.

Table 1.

First and Second Stage Indicators.

Phase of Analysis	Indicators
Phase 1 (efficiency evaluation)	Factors of production	Outcome
	Operating expenses, net worth	Aggregate income = Premium collected (net) + Income generated from investment
Phase 2 (evaluation of the influence of exogenous variables)	Solvency indicator ratio, Herfindahl–Hirschman (HH) concentration index, insurer age, and return on shareholders’ funds

Choice of Estimation Period

The current study includes the period 2012–2013 to 2019–2020. The study includes 16 (diversified) general insurance companies and four specialized health insurers functioning in India. The 16 diversified insurers include four public sector and 12 private sector insurers. All of the health insurers are privately owned. The data were procured from two sources: Insurance Regulatory and Development Authority (IRDA) Annual Reports for the corresponding years and the Indian Insurance Statistics Handbook for 2012–2013 through 2019–2020.

Results and Discussion

Testing for Convexity of the Production Technology

We have tested the convexity of the technology by using methods suggested by Kneip et al. (2016) and Simar and Wilson (2020). The maintained hypothesis of convexity of the technology set is tested against the alternative hypothesis of non-convexity of the technology set. This is accomplished by randomly splitting the sample into two subsamples. Then we have compared the mean of the DEA efficiency estimate from the first subsample (which follows convexity and variable RTS) with the mean of the free disposal hull (FDH) efficiency estimate derived from the second subsample (based on non-convexity). Further, bootstrap replications are made for eliminating the problem of sensitivity. Two tests have been performed: the mean of Kneip et al. (2016) test statistic over the sample splits and a Kolmogorov–Smirnov one-sample test for checking whether the split values obtained from the application of Kneip et al. (2016) test are uniformly distributed. The results obtained (Table 2) do not provide any indication to reject the null hypothesis.

Table 2.

Tests for Convexity of the Technology.

Test Name	Test Statistic	Value	p Value
Test statistic of Kneip et al. (2016)	Tau	1.2421	.19
One-sided Kolmogorov–Smirnov test	Tau	0.3599	.49

Summary Statistical Indicators of Efficiency Scores

Table 3 presents the mean efficiency scores of the in-sample insurance companies during the period under consideration (2012–2013 to 2019–2020). As mentioned earlier, we have used a global frontier approach for efficiency evaluation, that is, the input and output data for the whole 7-year period have been included in the reference set against which the efficiency performance of the in-sample insurers is estimated. Because of the use of a global frontier, efficiency scores are comparable across the observed years. Table 4 indicates that the mean efficiency shows a declining trend between 2012–2013 and 2016–2017, and improved thereafter in the next 3 years. The insurer-specific efficiency scores are included in Appendix Table A1.

Table 3.

Technical Efficiency Statistics of the In-Sample Insurers.

Year	Mean Efficiency	Standard Deviation	Skewness	Kurtosis
2012–2013	0.7825	0.1222	0.0260	−0.5752
2013–2014	0.7817	0.1059	0.4765	0.2932
2014–2015	0.7698	0.1179	1.2628	−0.0036
2015–2016	0.7461	0.1062	1.5156	1.1488
2016–2017	0.7279	0.0658	1.5533	1.7023
2017–2018	0.7385	0.0868	1.9339	3.4903
2018–2019	0.7579	0.1351	0.8102	−0.1754
2019–2020	0.8339	0.0832	0.8525	−0.2515

Table 4 (see Table A2 for insurer-wise bias-corrected efficiency scores) presents the outcomes of bootstrap technical efficiency that have been generated by applying the smoothed bootstrap (Simar & Wilson, 1998) with 200 replications. The table includes results relating to the mean of bootstrap technical efficiency with lower and upper bounds and bias-corrected technical efficiency.

Table 4.

Bootstrap Technical Efficiency of the In-Sample Insurers.

Statistic	2013	2014	2015	2016	2017	2018	2019	2020
Average lower bound	0.6888	0.6975	0.6823	0.6713	0.6563	0.6603	0.6797	0.7439
Average upper bound	0.7751	0.7735	0.7613	0.7389	0.7219	0.7315	0.7508	0.8266
Bias-corrected technical efficiency	0.7285	0.7333	0.7192	0.7055	0.6896	0.6953	0.7150	0.7838

Variations in Mean Technical Efficiency Across Insurer Types

Table 5 provides a comparison of the efficiency performance of diversified and specialized non-life insurers for the period under observation (2012–2013 to 2019–2020). During the period, movements in efficiency scores across the two categories of insurers are drastically different. For the diversified insurers, there is a clear trend of declining mean efficiency during the period. On the other hand, the mean efficiency scores for the health insurers are relatively stable.

Table 5.

Mean Technical Efficiency Performance Across General Insurer Categories.

Category	2013	2014	2015	2016	2017	2018	2019	2020
Private diversified	0.7527	0.7863	0.7325	0.7270	0.7100	0.7046	0.7328	0.8181
Private health	0.8131	0.6908	0.8605	0.8264	0.7901	0.7875	0.9462	0.8557
Private insurers	0.7678	0.7624	0.7645	0.7519	0.7300	0.7253	0.7862	0.8275
Public diversified	0.8415	0.8589	0.7912	0.7228	0.7195	0.7915	0.6446	0.8591

Ranking of General Insurers Based on Estimates of Technical Efficiency

Table 6 provides the ranking of the general insurers based on point and bias-corrected estimates of technical efficiency. Based on deterministic point estimates, the top four spots are occupied by Shri Ram General, New India Assurance, Care Health and Max Bupa, respectively. On the other hand, based on bias-corrected estimates, the top four spots are also occupied by the same insurers (albeit with different rankings). Bharti Axa and Reliance got the bottom ranks in both cases.

Table 6.

Ranking of General Insurers Based on Technical Efficiency Scores.

Insurer	Technical Efficiency	Rank	Insurer	Bias-corrected Efficiency	Rank
Shri Ram General	0.92	1	Max Bupa	0.82	1
New India	0.91	2	Care Health	0.81	2
Care Health	0.91	3	Shri Ram General	0.81	3
Max Bupa	0.90	4	New India	0.78	4
IFFCO Tokio	0.79	5	IFFCO Tokio	0.75	5
Royal Sundaram	0.76	6	Royal Sundaram	0.73	6
Bajaj Allianz	0.75	7	Bajaj Allianz	0.72	7
Star Health	0.75	8	Cholamandalam	0.72	8
Cholamandalam	0.75	9	Tata AIG	0.71	9
Tata AIG	0.74	10	Apollo Munich	0.71	10
Oriental	0.74	11	Star Health	0.71	11
Apollo Munich	0.74	12	ICICI Lombard	0.70	12
National	0.74	13	Oriental	0.70	13
United	0.73	14	SBI General	0.69	14
SBI General	0.72	15	Future Generali	0.69	15
ICICI Lombard	0.72	16	HDFC Ergo	0.69	16
Future Generali	0.71	17	National	0.68	17
HDFC Ergo	0.71	18	United	0.68	18
Bharti Axa	0.69	19	Bharti Axa	0.66	19
Reliance	0.69	20	Reliance	0.66	20

Estimation of RTS, Scale Elasticity and Efficiency

We have also estimated the RTS, scale efficiency and scale elasticity of the in-sample insurers. The results are included in Tables 7–9 and Tables A3–A5.

Table 8 presents summary information about RTS. We have considered year-specific DMU as separate entities. The table reveals that most of the DMUs have exhibited decreasing RTS. Thus, the number of DMUs exhibiting decreasing RTS varied between 16 (in 2019–2020) and 10 (2013–2014). The number of DMUs exhibiting IRS fluctuated between nine (maximum) and three (minimum). Very few insurer companies (New India Assurance for 2013–2014, Shri Ram General for 2013–2014 and Care Health Insurance for the remaining six in-sample years) exhibited constant RTS.

Table 7.

Returns to Scale Composition of the Observed General Insurers.

Particulars	2013	2014	2015	2016	2017	2018	2019	2020	Total
Decreasing returns to scale	12	10	11	11	14	14	15	16	103
Increasing returns to scale	7	9	8	8	5	5	4	3	49
Global (constant) returns to scale	1	1	1	1	1	1	1	1	8

Table 8 provides the descriptive statistics of scale efficiency for the in-sample period. The table shows that the mean technical efficiency improved between 2012–2013 and 2016–2017 and declined thereafter. Throughout the period under consideration, skewness remains negative, while negative kurtosis has been observed for 3 out of 7 years.

Table 8.

Descriptive Statistics of Scale Efficiency.

Particulars	2012–2013	2013–2014	2014–2015	2015–2016	2016–2017	2017–2018	2018–2019
Mean	0.8814	0.9365	0.9450	0.9454	0.9555	0.9480	0.9257
Standard deviation	0.1025	0.0617	0.0557	0.0768	0.0639	0.0781	0.0884
Skewness	−0.4135	−1.1435	−0.8632	−1.0908	−1.4378	−2.2179	−2.508
Kurtosis	−0.9398	0.4001	−0.5846	−0.6971	0.7941	5.0224	7.8742

Table 9 provides the descriptive statistical measures of the scale elasticity of output, which has been calculated (in the present case) as the negative of the input elasticity of the output distance function. The table indicates low values of (mean) scale elasticity for 2012–2013, 2016–2017, 2018–2019 and 2019–2020, as the values remain below unity. For the remaining 4 years under observation, the mean scale elasticity remains above unity. For all the years, the standard deviation of scale elasticity remained below 0.4. Skewness remained negative for 2019–2020 but positive for the remaining 7 years. Kurtosis is found to be negative for 2012–2013 through 2014–2015 and positive for the remaining 5 years.

Table 9.

Descriptive Statistics of Scale Elasticity.

Year	Average Efficiency	Standard Deviation	Skewness	Kurtosis
2012–2013	0.9273	0.2324	0.5206	−0.6250
2013–2014	1.0242	0.2225	0.4231	−0.8669
2014–2015	1.0638	0.2740	0.4144	−1.0052
2015–2016	1.0663	0.2439	1.1360	0.9665
2016–2017	0.9834	0.2087	1.0056	1.0548
2017–2018	1.0415	0.3777	2.8696	9.9828
2018–2019	0.8958	0.1927	1.1272	0.4923
2019–2020	0.8611	0.1869	−0.0697	0.1283

Second Stage Analysis: Regression of Efficiency Scores on the Contextual Variables

In the second stage of our study, we have regressed the efficiency (technical and scale) estimates on several contextual variables, including the solvency indicator (Herfindahl–Hirschman index (HHI) of market concentration), return on shareholders’ capital and insurer age. The results are presented in Tables 10 and 11. Initially, both pooled ordinary least squares (OLS) and (fixed and random effects) panel data models have been considered for explaining technical and scale efficiency. However, the comparison of the models on the basis of Hausman’s test indicated that for explaining technical efficiency, the fixed effects approach is found to be more suitable than the random effects model. On the other hand, the random effects approach is more appropriate for explaining scale efficiency.

Table 10 presents the regression results relative to the regression of technical efficiency on the contextual factors. The table shows that the coefficients of the two contextual variables (insurer age, return on equity) are statistically significant (at the 95% level of confidence). The table further indicates that efficiency is negatively related to insurer age and positively related to the return on equity. Technical efficiency is negatively related to the remaining two variables (index of market concentration and solvency ratio), and the coefficients are significant at the 90% level of confidence.

Table 10.

Fixed Effects Regression of Technical Efficiency on Contextual Variables.

Particulars	Coefficient (β)	Std Error (𝜎)	Coefficient/Std Error	p Value
Intercept	.7466	0.3840	1.944	.0668
Return on shareholders’ capital	.0337	0.0419	0.8043	.4312
Solvency ratio	−.0265	0.0188	−1.406	.1759
Insurer age	−.0189	0.01003	−1.881	.0754
Herfindahl–Hirschman index	−6.7973	2.36553	−2.873	.0097

Table 11 presents the regression results of scale efficiency estimates on the same contextual variables. The results indicate that the coefficients of HHI and insurer age are statistically significant. However, scale efficiency is positively linked to insurer age. The coefficient of return on equity is positive, and that of the solvency ratio is negative.

Table 11.

Random Effects Regression of Scale Efficiency on Contextual Variables.

Particulars	Coefficient	Std Error	z	p Value
Intercept	−1.1280	0.4029	−2.800	.0114
Return on shareholders’ capital	0.0052	0.0067	0.7824	.4437
Solvency ratio	−0.1246	0.0657	−1.896	.0733
Insurer age	0.0257	0.0086	2.967	.0079
Herfindahl–Hirschman index	6.3209	2.7761	2.277	.0345

Concluding Remarks

The current study finds a number of important results by reinvestigating the trends in efficiency and scale properties of Indian general insurance and exploring the impact of contextual variables on technical and scale efficiency. First, both deterministic and bias-corrected technical efficiency exhibit an initial decline followed by a reversal of the trend by the end of the phase under observation. The changes in year-on-year scale efficiency have been less dramatic. Further, while the z-index of concentration and insurer age remained as the two most influential contextual variables, their impact on technical and scale efficiency has been diametrically opposite. Finally, most of the insurers exhibited decreasing RTS and low values of scale elasticity, which is worrisome.

The study, however, has a few shortcomings. First, the sample size is small, limited to 20 major diversified insurance companies. In recent times, a number of new insurers have penetrated the market, and the impact of the changed scenario needs investigation. Further, the study covers an 8-year span only and needs a separate investigation for the recent years. The assumption of global convexity also needs more detailed investigation, and in the case of non-convexity, a clustering approach to DEA (Sinha & Amirteimoori, 2025) might be a better option. Future research studies on Indian insurance companies can take care of the aforementioned limitations.

Footnotes

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Ram Pratap Sinha

Appendix A

References

Alhassan

A. L.

, & Biekpe

(2015). Efficiency, productivity and returns to scale economies in the non-life insurance market in South Africa. The Geneva Papers on Risk and Insurance—Issues and Practice , 40(3), 493–515.

Banker

R. D.

(1984 ). Estimating most productive scale size using data envelopment analysis. European Journal of Operational Research , 17(1), 35–44. https://doi.org/10.1016/0377-2217(84)90006-7

Banker

R. D.

, Charnes

, & Cooper

W. W.

(1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science , 30(9), 1078–1092.

Banker

R. D.

, & Natarajan

(2008). Evaluating contextual variables affecting productivity using data envelopment analysis. Operations Research , 56(1), 48–58.

Banker

R. D.

, Natarajan

, & Zhang

(2019). Two-stage estimation of the impact of contextual variables in stochastic frontier production function models using data envelopment analysis: Second stage OLS versus bootstrap approaches. European Journal of Operational Research , 278(2), 368–384.

Banker

R. D.

, & Thrall

R. M.

(1992). Estimation of returns to scale using data envelopment analysis. European Journal of Operational Research , 62(1), 74–84, https://doi.org/10.1016/0377-2217(92)90178-C

Barros

C. P.

, Nektarios

, & Assaf

(2010). Efficiency in the Greek insurance industry. European Journal of Operational Research , 205(2), 431–436.

Berger

A. N.

, Hanweck

, & Humphrey

(1987). Competitive viability in banking: Scale, scope and product mix economies. Journal of Monetary Economics , 20(3), 501–520.

Berger

A. N.

, & Humphrey

(1992). Measurement and efficiency issues in commercial banking. In

Griliches

(Ed.), Output measurement in the service sectors (pp. 245–300). University of Chicago Press.

10.

Charnes

, Cooper

W. W.

, & Rhodes

(1978). Measuring the efficiency of decision-making units. European Journal of Operational Research , 2(6), 429–444.

11.

Choi

B. P.

, & Weiss

M. A.

(2005). An empirical investigation of market structure, efficiency, and performance in property-liability insurance. Journal of Risk and Insurance , 72, 635–673. https://doi.org/10.1111/j.1539-6975.2005.00142.x

12.

Cummins

D. J.

, & Weiss

M. A.

(1993). Measuring cost efficiency in the property-liability insurance. Journal of Banking and Finance , 17, 463–481.

13.

Cummins

J. D.

, & Xie

(2013). Efficiency, productivity and scale economies in the U.S. property-liability insurance industry. Journal of Productivity Analysis , 39(2), 141–164.

14.

Eling

, & Luhnen

(2010). Frontier efficiency methodologies to measure performance in the insurance industry: Overview, systematization, and recent developments. The Geneva Papers on Risk and Insurance—Issues and Practice , 35(2), 217–265.

15.

Ennsfellner

K. C.

, Lewis

, & Anderson

R. I.

(2004). Production efficiency in the Austrian insurance industry: A Bayesian examination. Journal of Risk Insurance , 71(1), 135–159.

16.

Färe

, Grosskopf

, & Lovell

C. A. K.

(1988). Scale elasticity and scale efficiency. Journal of Institutional and Theoretical Economics (JITE) , 144, 721–729 http://www.jstor.org/stable/40751123.

17.

Farrell

M. J.

(1957). The measurement of productive efficiency. Journal of Royal Statistical Society Series A , 120(3), 253–281.

18.

Fenchel

(1949). On conjugate convex functions. Canadian Journal of Mathematics , 1(1), 73–77. https://doi.org/10.4153/CJM-1949-007-x

19.

Ferro

, & León

S. A.

(2017). Stochastic frontier analysis of efficiency in Argentina’s non-life insurance market. The Geneva Papers on Risk and Insurance—Issues and Practice , 43(1), 158–174.

20.

Fukuyama

, & Weber

W. L.

(2001). Efficiency and productivity change of non-life insurance companies in Japan. Pacific Economic Review , 6(1), 129–146. https://doi.org/10.1111/1468-0106.00122

21.

Hancock

(1985). The financial firm: Production with monetary and nonmonetary goods. Journal of Political Economy , 93(5), 859–880.

22.

Hancock

(1991). A theory of production for the financial firm. Kluwer Academic Publishers.

23.

Ilyas

A. M.

, & Rajasekharan

(2019a). An empirical investigation of productivity and efficiency in the Indian non-life insurance market. Benchmarking: An International Journal . https://doi.org/10.1108/BIJ-01-2019-0039

24.

Ilyas

A. M.

, & Rajasekharan

(2019b). Productivity of the Indian non-life insurance sector: New evidence using Fare-Primont index approach. International Journal of Productivity and Performance Management , 69(4), 633–650. https://doi.org/10.1108/IJPPM-04-2019-0147

25.

Kao

, & Hwang

S. N.

(2008). Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan. European Journal of Operational Research , 185(1), 418–429.

26.

Kneip

, Simar

, & Wilson

P. W.

(2016). Testing hypothesis in nonparametric models of production. Journal of Business and Economic Statistics , 34, 435–456.

27.

Kramarić

T. P.

, Pervan

, & Ćurak

(2022). Determinants of Croatian non-life insurance companies’ efficiency. Croatian Operational Research Review , 13(2), 149–160.

28.

Leverty

J. T.

, & Grace

M. F.

(2010). The robustness of output measures in property-liability insurance efficiency studies. Journal of Banking and Finance , 34(7), 1510–1524.

29.

Mahlberg

, & Url

(2010). Single market effects on productivity in the German insurance industry. Journal of Banking & Finance , 34(7), 1540–1548.

30.

Mandal

, & Dastidar

S. G.

(2014). A DEA-investigation of efficiency of the Indian insurance during recession. Journal of Advances in Management Research , 11(1), 115–136.

31.

Panzar

J. C.

, & Willig

R. D.

(1977). Economies of scale in multi-output production. The Quarterly Journal of Economics , 91(3), 481–493. https://doi.org/10.2307/1885979

32.

Rai

(1996). Cost efficiency of international insurance firms. Journal of Financial Services Research , 10(3), 213–233.

33.

Sealey

C. W.

, & Lindley

J. T.

(1977). Inputs, outputs, and a theory of production and cost at depository financial institutions. The Journal of Finance , 32(4), 1251–1266. https://doi.org/10.2307/2326527

34.

Shephard

R. W.

(1953). Cost and productions . Princeton University Press.

35.

Shephard

R. W.

(1970). Theory of cost and productions . Princeton University Press.

36.

Shephard

R. W.

(1974). Indirect production functions . Verlag Anton Hain.

37.

Simar

, & Wilson

PW.

(1998). Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science , 44(1), 49–61.

38.

Simar

, & Wilson

P. W.

(2020). Hypothesis testing in nonparametric models of production using multiple sample splits. Journal of Productivity Analysis , 53, 287–303.

39.

Sinha

R. P.

(2007). Productivity and efficiency of Indian general insurance industry. ICFAI Journal of Risk & Insurance , 4(2), 33–43.

40.

Sinha

R. P.

(2017a). Dynamic performance benchmarking of Indian insurance companies. Journal of Quantitative Economics , 15(3), 543–564.

41.

Sinha

R. P.

(2017b). Efficiency-solvency linkage of Indian insurance companies: A robust non-parametric approach. Eurasian Economic Review , 7(3), 353–370.

42.

Sinha

R. P.

, & Amirteimoori

(2025). Efficiency of Indian life insurance companies. In

Emrouznejad A., Ray

, & Patel

G. N.

(Eds.), Advances on the theory and practice of data envelopment analysis . Springer.

43.

Toivanen

(1997). Economies of scale and scope in the Finnish non-life insurance industry. Journal of Banking & Finance , 21(6), 759–779.

44.

Weiss

M. A.

(1991). Efficiency in the property-liability insurance. Journal of Risk and Insurance , 58(3), 452–479 https://doi.org/10.2307/253402.

45.

Yang

(2006). A two-stage DEA model to evaluate the overall performance of Canadian life and health insurance companies. Mathematical and Computer Modelling , 43(7–8), 910–919.

Efficiency and Scale Properties of Indian General Insurers—A DEA Approach

Abstract

Keywords

Introduction

An Inspection of the Prior Studies

International Studies

Indian Studies

Motivating Factors Behind the Current Research

Methodology

Technical Efficiency

Scale Efficiency

Formal Presentation of the Evaluation Methodology

Estimation of Scale Elasticity

Relationship of Efficiency with Explanatory Variables

The Evaluation Framework

Choice of Input, Output and Exogenous Variables

First and Second Stage Indicators.

Choice of Estimation Period

Results and Discussion

Testing for Convexity of the Production Technology

Tests for Convexity of the Technology.

Summary Statistical Indicators of Efficiency Scores

Technical Efficiency Statistics of the In-Sample Insurers.

Bootstrap Technical Efficiency of the In-Sample Insurers.

Variations in Mean Technical Efficiency Across Insurer Types

Mean Technical Efficiency Performance Across General Insurer Categories.

Ranking of General Insurers Based on Estimates of Technical Efficiency

Ranking of General Insurers Based on Technical Efficiency Scores.

Estimation of RTS, Scale Elasticity and Efficiency

Returns to Scale Composition of the Observed General Insurers.

Descriptive Statistics of Scale Efficiency.

Descriptive Statistics of Scale Elasticity.

Second Stage Analysis: Regression of Efficiency Scores on the Contextual Variables

Fixed Effects Regression of Technical Efficiency on Contextual Variables.

Random Effects Regression of Scale Efficiency on Contextual Variables.

Concluding Remarks

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

Appendix A

References