Estimating Revenue Efficiency in Indian General Insurance Companies: A Semi-parametric Econometric Approach
Abstract:
In the realm of general insurance in India, an econometric investigation was conducted to estimate the revenue efficiency across a selection of 15 prominent, diversified general insurance entities for the fiscal years 2011-12 to 2016-17. Utilizing a semi-parametric methodology, the revenue frontier was constructed under the GAM framework, while the variance components were estimated employing the method of moments. This analysis further explored the influence of revenue efficiency on critical profitability metrics, namely return on equity (ROE) and return on assets (ROA), through the application of instrumental variable regression. The findings provide pivotal insights into the dynamics of revenue efficiency and its consequential impact on the financial performance of general insurance companies in India, offering a substantial contribution to the literature on insurance economics and the methodology of efficiency measurement. The research underscores the significance of adopting semi-parametric models for a nuanced understanding of revenue efficiency, thus paving the way for enhanced strategic decision-making in the insurance sector.1. Introduction
The general insurance industry plays a crucial protective role in any modern society as it compensates for losses in property, liability and health of the insured. General insurance includes all forms of insurance except life insurance. The principal forms of general insurance include: (i) insurance of property (against incidence of theft, burglary or fire); (ii) insurance of health (against physical illness); and (iii) insurance against liability (professional, product and public liabilities).
In the context of Indian insurance market, the segment of general insurance compared to the total insurance market is relatively miniscule; general insurance accounts for around 22% of the total insurance market in India. However, the industry has tremendous potential for expansion in view of the growing need to protect properties and agricultural produce from natural calamities (or incidents like a fire breakout), protect businesses and individuals against contingent liabilities and provide a cushion against rising health care costs.
In the post-reform phase, the efficiency and productivity performance of the general insurance companies have been evaluated by several studies using aDEA (non-parametric) framework and most of them attempted to evaluate the economic/technical efficiency of the general insurers. The present study, in contrast, adopted a different approach. In the first phase, a revenue frontier of the in-sample general insurers has been constructed utilising a semi-parametric General Additive Model-Stochastic Frontier Analysis (GAM-SFA) approach wherein the frontier is estimated by applying the GAM and the variance parameters are obtained by using pseudo-likelihood estimators. In the second stage, the relationship between important profitability parameters and revenue efficiency estimates is explored in terms of regression with an instrumental variable. The study has five sections and proceeds as follows: Section 1 provides a brief overview of the Indian general insurance industry. Section 2 reviews the extant literature on general insurance performance. Section 3 outlines the methodology. Section 4 relates to the data, results and discussion. Section 5 is the conclusion.
2. General Insurance Industry in India
The origin of the general industry sector in India is quite old. The contemporary form of general insurance industry originated in India much before independence. Thus, Triton Insurance Co. Ltd. Came in to existence in 1850. This was succeeded by the formation of the Bombay Mutual Life Insurance Society (in 1870) and the Oriental Assurance Company (in 1880). Thereafter, numerous non-life insurance companies were established in India. Consequently, more than 100 non-life insurance companies existed in India at the juncture of the nationalization (1973) when the industry came under government control. The process of government take over (of the industry) involved the formation of the General Insurance Corporation as the government-sponsored reinsurance entity and the establishment of (four) government-sponsored general insurance companies (by merging 107 non-government non-life insurers).
In the 1990s, India embraced an open market economy with a greater role for the private sector. Financial sector reform was an integral part of the reform process which was initiated with banking sector reform involving the deregulation of private sector entry and the introduction of prudential regulations for banking operations. At the same time, the Government of India set up a high-powered committee on the Indian insurance sector (headed by Shri R. N. Malhotra) to examine the existing scenario and recommend appropriate measures for boosting competitive efficiency and strengthening the regulatory framework. The Committee recommended deregulation in both the life and general insurance markets. Private sector players started entering the general insurance market in 2000 and by March 2017, the total number of general insurance companies had increased to 31. During the post-liberalisation phase, the sector witnessed significant growth in terms of premium growth, asset management and the branch network of the insurance companies. Table 1 provides a brief overview of the growth observed in the sector between end-March 2012 and end-March 2017. Table 2 provides the movements in insurance penetration (ratio of aggregate insurance premium to GDP), insurance density (ratio of insurance premium to total population) and incurred claims ratio (ratio of total paid claims to total premium collected) during the same period.
Particulars | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Number of general insurance companies (including reinsurers) | 28 | 28 | 29 | 29 | 30 | 31 |
Number of offices | 7050 | 8099 | 9,872 | 10407 | 10803 | 11141 |
Number of new non-life insurance policies issued (in lakhs) | 857 | 1070 | 1048 | 1202 | 1257 | 1525 |
Gross direct premium (Rs crores) | 54578 | 65023 | 79934 | 87151 | 99333 | 130971 |
Assetsunder management (Rs crores) | 99268 | 122992 | 149536 | 172144 | 188126 | 222344 |
Particulars | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Insurance penetration | 0.78 | 0.8 | 0.7 | 0.7 | 0.70 | 0.77 |
Insurance density | 10.5 | 11 | 11 | 11 | 11.5 | 13.2 |
Incurred claim ratio | 88.89 | 82.79 | 76.54 | 73.58 | 74.44 | 84.38 |
3. Related Work
The present section briefly outlines the important research studies related to general insurance companies, both in the international and Indian contexts. Section 2.1 provides a brief summary of the international research studies while Section 2.2 describes the Indian studies.
Toivanen [1] estimated the economies of scale and scope for the Finnish non-life insurance industry for the period 1989-91. The study identified two types of costs (operating costs and costs arising from incurred claims) related to production and portfolio management functions. The study emphasized on the formation of a branch network for gaining market power or information acquisition benefits since it helps captivate customers from the standpoint of portfolio management. The study established the existence of scale diseconomies among the firms, scale economies at the branch level and scope economies in production. Fukuyama and Weber [2] estimated the efficiency and productivity growth of Japanese non-life insurance companies for the period 1983-1994. They estimated output-oriented technical efficiency and the Malmquist index of total factor productivity change. The productivity change estimate was further decomposed into efficiency change and frontier shift components. The study found significant productivity improvement during 1983-90 which was primarily contributed by technical change. In the succeeding three years, technical change stagnated (due to the collapse of the bubble economy) but started improving again by 1993-94.
Ennsfellner et al. [3] examined the production efficiency of the Austrian insurance market for the time period 1994-1999. The study used a Bayesian stochastic frontier for estimating the aggregate and firm-specific efficiency of the in-sample insurance companies. The study found that the process of deregulation of the Austrian insurance market had influenced the productive efficiency of the insurers in a positive manner.
Choi and Weiss [4] examined the performance of the US property-liability industry using a stochastic frontier approach.In particular, the study examined the linkage between firm performance, market structure and efficiency in property-liability insurers during 1992–1998.The study estimated both cost and revenue efficiency and examined the soundness of three following hypotheses: traditional SCP(Structure-Conduct-Performance),RMP( the Relative Market Power), and ES (Efficient Structure). The results obtained from the aforementioned study provided support for the efficient structure hypothesis. According to the efficient structure hypothesis, more efficient firms are able to charge lower prices than competitors. This permits them to get larger market shares and extract economic rents, resulting in increased concentration in the industry. Thus, as per the study, insurance regulators should be more concerned about efficiency compared to relative market power.
Yang [5] applied a two-stage non-parametric model for evaluating the efficiency of the Canadian life and health insurance industry. His study includes two separate models for the computation of production and investment efficiency for both input- and output-oriented models. In the second stage of the study, a dummy input with value 1 was used and the production and (inverse of) investment efficiency scores were used as outputs. This model thus involved the integration of production and investment performance for the observed insurance companies. The evidence obtained from the study indicated that the life and health insurance industry in Canada operated fairly efficiently during the in-sample period.
Kao and Hwang [6] applied a relational and independent two-stage Data Envelopment Analysis (DEA) model to evaluate the performance of 24 non-life insurance companies in Taiwan using average data for 2001 and 2002. The study decomposed the activity of insurance firms into two sub-processes: marketing (intermediate stage) and investment (final stage). The study showed that efficiency in the investment sub-process was lower than in the first-stage activity (marketing). Cummins and Xie [7] studied the effects of mergers and acquisitions in the US property-liability insurance industry on the productivity and efficiency of the insurers using data from 1994–2003. The results suggested that mergers and acquisitions in property-liability insurance increased the valuation of the concerned insurers. Their study found that acquiring insurers achieved more revenue efficiency gains than non-acquiring companies, and target firms exhibited greater cost and allocative efficiency growth as compared to the companies which were not targeted.
Barros et al. [8] used two-stage conditional performance benchmarking for the efficiency evaluation of 71 Greek life and non-life insurance companies. The study assumed that the returns to scale were global for 1994 to 2003. The first-phase outcomes of the study revealed significant divergence in efficiency performance for the in-sample time span. The second-phase (regression) results indicate that while extent of competition is a major influential variable of efficiency in the Greek insurance industry, this was not enough to accelerate market efficiency for the observed period. Mahlberg and Url [9] examined the efficiency and productivity impact of market unification on the German insurance companies for the period 1991-2006. Their study estimated the technical, revenue, cost efficiency and total factor productivity of 202 insurance companies under non-increasing returns to scale. The study provided a mixed picture regarding the convergence of performance among the observed insurance companies.
Cummins and Xie [10] examined efficiency, productivity and scale economies in the US property-liability insurance industry. The study analysed efficiency and change in total factor productivity using DEA. The results showed that most of the insurers below median size in the industry exhibited IRS (increasing returns to scale), and the majority of the property-liability insurers above median size exhibited DRS (decreasing returns to scale). Jarraya and Bouri [11] investigated profit efficiency and optimal production targets for the European non-life insurance industry for the period 2002-2008.They used the directional distance function for the computation of profit efficiency and the Lagrangean function for the specification of an optimal production plan. The study estimated the parameters of a stochastic technology frontier constructed from a sample of 175 non-life insurance companies. On the basis of the optimal quantity of inputs, good outputs and bad outputs generated from their model, the study proposed indices of inefficiency for profit and production factors.
Alhassan and Biekpe [12] estimated the efficiency, productivity and returns to scale of South African non-life insurance companies for the period 2007-2012. They employed DEA for estimating efficiency, returns to scale and Malmquist total factor productivity change for the aforementioned period. They applied truncated bootstrapped and logistic regression techniques for identifying the determinants of efficiency and the probability of operating under constant returns to scale. The results showed that non-life insurers operated with about 50 percent efficiency. Approximately 20 percent of insurers were scale-efficient. The study also found productivity improvements during the period which were mainly due to technological changes. The results of the regression analysis indicated a non-linear impact of size on efficiency and constant returns to scale. Variables like product line diversification, reinsurance and leverage also had a significant relationship with efficiency and constant returns to scale.
Ferro and Leon [13] applied SFA to estimate the technical efficiency of Argentine non-life insurance companies for the period 2009-2014. The study applied two models: a time-invariant inefficiency model and time-varying decay model. The results indicated a low average of technical efficiency, a stagnated efficiency level during the later phase of the observed time period and a negative technical change.
Mandal and Ghosh [14] estimated the technical efficiency of 12 general insurance companies in India for the period 2006-07 to 2009-10 using DEA. The study sought to assess the impact of the global economic slowdown on the general insurance sector. Sinha [15] used a dynamic DEA framework for estimating the dynamic efficiency performance of Indian general insurance companies. In a second study, Sinha [16] estimated the efficiency of Indian general insurance companies for 2013-14 using the conditional performance benchmarking method. Ilyas and Rajasekharan [17], [18] estimated the efficiency, total factor productivity and returns to scale of the Indian non-life insurance industry over the period 2005-2016. The first study [17] found that the Indian non-life insurance industry was moderately technical, scaled, cost-effective, and allocatively efficient. The second study [18] found (using the Fare-Primont index) that the non-life insurance sector exhibited a very low level of total factor productivity. The total factor productivity growth observed during the observed period (2005-2016) is mainly attributable to scale-mix efficiency.
The present study takes a different approach in the estimation of the efficiency performance of the in-sample general insurance companies in two respects. First, it seeks to estimate the revenue efficiency of the observed general insurers which has not been attempted so far in the existing literature. Second, the current study adopts a relatively novel approach to the estimation of efficiency.
4. Methodology
This section outlines the methodological development related to the estimation of stochastic frontiers and also outlines the regression method applied in the second stage.
Neoclassical economics defines a production function as one which relates the maximum producible level of output to a fixed bundle of inputs. Thus, for practical purposes, one can think of an industry production function which implies a frontier of potential output corresponding to the given input level while individual production functions may vary from the frontier because of variations in the scale of operation or heterogeneity in organizational structure. However, in the conventional econometric approach, the average production function was estimated as the representative of the industry production function. Farrell [19] defined the technical efficiency of an industry with respect to a given efficient isoquant as the weighted average of the technical efficiency of the corresponding isoquants of the constituent firms. A major weakness of this average approach is that the performance of a firm is compared with the average performance and not the best practice firm. Initial efforts were made by Aigner and Chu [20] using linear and quadratic programming with single equations and two-stage least squares techniques. For a production function $Y=f(X ; \beta)$, $Y$ is the maximum producible output from $X$ (a non-stochastic input vector), and $\beta$ represents the unknown vector of parameters to be estimated. The authors suggested two alternative minimization programs: (i) $\operatorname{Min} [ Y_i-f\left(X_i ; \beta\right)]$ subject to $Y_i \leq f[\left(X_i ; \beta\right)]$, and (ii) $\operatorname{Min}\left[Y_i-f\left(X_i ; \beta\right)\right]^2$ subject to $Y_i \leq f\left(X_i ; \beta\right)$. However, the results obtained were highly sensitive to the outliers [21]. Therefore, the econometric model of $Y_i=f\left(X_i ; \beta\right)+\epsilon_i$ was considered.
The error term can be decomposed into $\epsilon_i=u_i+v_i$, where $v_i$ represents a symmetric disturbance term which exhibits an independent and identical distribution (iid) as $N\left(0, \sigma_v^2\right), u_i$ represents a one-sided disturbance term ($u_i \leq 0$) and is derived from a distribution ($0, \sigma_u^2$), which is truncated above 0. The residual of the estimate of the aforementioned model is $\hat{\epsilon}_i=Y_i-f\left(X_i ; \hat{\beta}\right)$. Meeusen and van den Broeck [22] introduced an efficiency model $Y_i=\theta\left(X_i\right) E_i U_i$,, where $E_i$ is a measure of efficiency distributed in the range (0,1), and $U_i$ is a disturbance term distributed in the range (0,$\alpha$).However, one needs to have separate estimates of $u_i$ and $v_i$. Jondrow et al. [23] provided a procedure for the estimation of individual efficiency scores for a particular point ($x,y$) on the production frontier. The total variance is decomposed into the sum total of variances of $u$ and $\sigma^2=\sigma_u^2+\sigma_v^2, u_*(\mathrm{x})=-\frac{\sigma_u^2 \epsilon}{\sigma^2}, \sigma_*^2=\frac{\sigma_u^2}{\sigma^2} \sigma_v^2$. For the conditional distribution of $u(u \mid \epsilon) \sim N\left(u_*, \sigma_*^2\right)$, the variable is truncated at 0. For getting a point estimate of technical inefficiency, either the mean or mode of the distribution can be used. In addition, the result $-\frac{\mu_*}{\sigma_*}=\frac{\epsilon \lambda}{\sigma}$ can be obtained with $\lambda=\frac{\sigma_u}{\sigma_v}$.
The initial contributions to the stochastic frontier literature considered cross-sectional models and thus did not consider the impact of technical inefficiency changes over time. Cornwell et al. [24] and Kumbhakar [25] introduced stochastic frontier models in which technical inefficiency changes over time. They considered that firm effects are a quadratic function of time with coefficients varying across firms with a multi-variate distribution. In the Kumbhakar model [25], technical inefficiency is assumed to be a product of a well-defined deterministic function of time and a non-negative time-invariant effect which varies across firms. Battese and Coelli [26] introduced a stochastic frontier model in which firm effects are an exponential function of time. Battese and Coelli [27] introduced a more general model in which technical inefficiency effects are a function of idiosyncratic variables and time.
Fan et al. [28] introduced a semi-parametric frontier model, in which the functional form of the production frontier is unspecified, and the distribution of the composed error terms isin the known form as follows:
$Y_i=g\left(X_i\right)+\epsilon_i\left(\epsilon_i=v_i-u_i\right)$
where, $X_i$ is a vector of regressors, $g(X_i)$ represents an unknown smooth function, $\epsilon_i$ is a composed error term including a two-sided statistical noise and a one-sided error term representing technical inefficiency.
The probability density function of $\epsilon$ is as follows:
$ f(\epsilon)=\frac{2}{\sigma} \varphi\left(\frac{\epsilon}{\sigma}\right)\left[1-\psi\left(\epsilon \lambda \sigma^{-1}\right)(-\alpha \leq \epsilon \leq \alpha)\right.$
where, $\sigma^2=\sigma_u^2+\sigma_v^2, \lambda=\frac{\sigma_u}{\sigma_v}$, and $\varphi(.)$ and $\psi(.)$ are the standard normal density and distribution functions, respectively.
The conventional stochastic frontier approach requires the specification of an explicit functional form for relating output to input levels. In order to overcome this lack of flexibility, Vidoli and Ferrara [29] introduced a GAM framework. GAMs were proposed by Hastie and Tibshirani [30] which extended the conventional likelihood-based regression models and introduced a method for estimation. In the case of General Linear Model (GLM), the linear model $Y_i=\beta_0+\sum_{i=1}^n \beta_i X_i$ is replaced with $\beta_0+\sum_{i=1}^n f_i\left(X_i\right)$, with $f_i(.)$ as the smooth function.
In the present study, the GAM-based SFA (stochastic frontier approach) is used to construct the revenue frontier and estimate efficiency. The variance parameters are estimated using Fan et al. [28].
An important part of the present study is to explore how the measures of revenue efficiency explain the profitability indicators of the in-sample general insurance companies. For this, an appropriate regression model has to be formulated with efficiency measured as the explanatory variable. However, revenue efficiency cannot be estimated directly. The estimate obtained depends significantly on the methodology being used. Further, the efficiency variable is stochastic in nature. In such cases, the instrumental variable method can be used, with the explanatory variable semi-replaced by another variable, which is correlated with the explanatory variable instead of the error term.
To explain the instrumental variable method in brief, the following regression model is considered:
$Y_i=\beta_0+\beta_1 X_i+U_i$
where, $Y_i, X_i$ and $U_i$ represent the explained variable, the explanatory variable and the error term, respectively.
Now another variable, $Z$, is considered, which is correlated with $X$ instead of the error term. The instrumental variable estimator of the slope $\left(\beta_1\right)$ may be defined as
$\bar{\beta}_1^{\bar{IV}}=\frac{\operatorname{Cov}(Y, X)}{\operatorname{Cov}(Z ; X)}$
where, $\bar{\beta}_1^{\bar{IV}}$ is a consistent estimator of the slope provided $Cov(Z. X)$ is non-zero. $\bar{\beta}_1^{\bar{IV}}$ is related to the true estimate of the slope in the form of $\bar{\beta}_1^{\bar{IV}}=\beta_1+\frac{Cov(Z, U)}{Cov(Z, X)}$. If $Cov(Z, U)=0$, then $\bar{\beta}_1^{\bar{IV}}\cong \beta_1$.
5. Data, Variables and Estimation Framework
Estimation of revenue frontiers and revenue efficiency requires the selection of input indicators, output indicators and (input and output) prices. However, specification of input and output variables (and price parameters) in the context of a general insurance industry is a difficult task because of the complexities involved in defining the indicator variables.
Eling and Luhnen [31] identified three major types of inputs used in the insurance industry: labour (including agents and office staff), business services (including items such as travel, communications and advertisement) and capital (including debt and equity capital). On the output side, Leverty and Grace [32] found three alternative approaches for choosing outputs: the Financial Intermediation (FI) approach, the User Cost (UC) approach and the Value Added (VA) approach. In the context of commercial banking and other financial service providers engaged in fund-mobilisation and allocative activities, this intermediation view considers financial service units as intermediation firms who bridge the gap between demanders and suppliers of funds. The value-added approach considers those activities as outputs which contribute significant value as assessed using operating cost allocations [33]. Broadly speaking, the value-added approach assumes that the insurers provide three major services: risk-pooling and risk-bearing, real financial services and financial intermediation. Some studies have used net premiums as valued added while others have used incurred benefits and changes in reserves as output proxies [34].
The present study seeks to construct a revenue frontier. Therefore, the major activities need to be identified, which contribute to insurer revenue as well as the inputs of service production. The problem in the Indian context, however, is the absence of very detailed information about variables which may be used as inputs and outputs for the estimation of revenue frontiers and revenue efficiency. Since very detailed information on various inputs cannot be obtained separately, two inputs are considered: total operating expenses and assets under management. The first input is considered as the proxy input for capturing operational activities which are required to generate premium. The second input captures the investment activities of general insurance companies. On the output side, net premium income and income from investments are considered as the two outputs. For all the output indicators, the corresponding prices are considered as unity. Table 3 presents a brief statement of input indicators, output indicators and prices.
Description | Input | Output |
Operating expenses | √ | - |
Assetsunder management | √ | - |
Cross-section dummy | √ | - |
Time series dummy | √ | - |
Total income=net premium income+investment income | - | √ |
The present study covers the period 2011-12 to 2016-17. The data relating to the indicator variables have been collected from two sources: the Annual Reports of the IRDA for the respective years and the Handbook of Indian Insurance Statistics for 2014-15 and 2016-17.
In the present study, two models of estimation of revenue frontiers are considered as follows:
$\begin{gathered}\operatorname{Ln}(Y)=f\left[\operatorname{Ln}\left(X_1\right), \operatorname{Ln}\left(X_2\right), \operatorname{Ln}\left(X_3\right), \operatorname{Ln}\left(X_4\right)\right] \\ \operatorname{Ln}(Y)=f\left(X_1, X_2, X_3, X_4\right)\end{gathered}$
In the first model (Model 1), logarithmic values of the output and input variables are considered. In the second model (Model 2), logarithmic values of only the output variable are considered. For both models, a two-step procedure is used for the construction of the frontier. In the first step, GAM is used for constructing the frontier and computing efficiency. In the second step, variance parameters are obtained by the procedure.
This section presents efficiency scores from the two models, parameter coefficients, goodness of fit, variance component estimates and linkage with two important measures of profitability (ROE and ROA).
Table 4 provides the Measures of location and dispersion of efficiency of efficiency scores for the two models estimated (presently) for the observation period (2011-12 to 2016-17). The mean efficiency scores for the two models revealed fluctuating trends during the observed time span. Thus, mean efficiency scores declined between 2011-12 and 2012-13, improved in the next financial year (2013-14) and declined for the next two years (2014-15 and 2015-16), In the last year under observation, mean scores improved somewhat in the first model while continuing the declining trend in the second model. The detailed results relating to revenue efficiency scores are included in Tables A1 and A2 in the appendix.
Model | Descriptive statistics | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Model 1 | Mean | 0.8070 | 0.7966 | 0.8404 | 0.8278 | 0.7815 | 0.7993 |
Standard deviation | 0.1533 | 0.0981 | 0.0553 | 0.0807 | 0.1041 | 0.0857 | |
Model 2 | Mean | 0.8210 | 0.8091 | 0.8267 | 0.8195 | 0.7885 | 0.7821 |
Standard deviation | 0.1406 | 0.0932 | 0.0450 | 0.0644 | 0.0951 | 0.0993 |
Information about how technical efficiency varies across public and private sector general insurers is obtained. Table 5 provides a comparison of the mean technical efficiency scores of observed public and private sector general insurers with respect to the two models.
Model | Category | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Model 1 | Public sector | 0.8348 | 0.7864 | 0.8057 | 0.7595 | 0.7974 | 0.8357 |
Private sector | 0.7969 | 0.8003 | 0.8531 | 0.8526 | 0.7758 | 0.7860 | |
Model 2 | Public sector | 0.8197 | 0.8216 | 0.8127 | 0.8175 | 0.8521 | 0.8309 |
Private sector | 0.8215 | 0.8045 | 0.8318 | 0.8202 | 0.7654 | 0.7643 |
Table 5 shows that there is not much difference in technical efficiency across ownership categories, irrespective of the model considered for estimation. The difference is even less if the average of the six in-sample years is taken. However, the relative performance of private and publicly owned general insurance companies varies across the two models chosen for estimation.
Based on the estimates derived from the application of GAM-SFA, efficiency density plots can be generated. The density plots estimated with respect to the two models (Model1 and Model 2) are presented in Figure 1. Figure 1(a) shows the efficiency density plot for Model 1 while Figure 1(b) presents the density plot for Model 2.
Stochastic frontier models include two error terms representing random (white noise) error ($v$) and technical inefficiency ($u$). Because of the presence of $v$, there are random deviations from the revenue frontier. On the other hand, it implies that there can be one-sided (negative) deviations from the boundary of the revenue frontier, which is defined by the best practices of insurers. Table 6 presents the estimates of technical inefficiency for the two models. Tables A3 and A4 in the Appendix present the year-wise and insurer-wise estimates.
Model | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Model 1 | 0.2419 | 0.2348 | 0.1759 | 0.1936 | 0.2553 | 0.2296 |
Model 2 | 0.2168 | 0.2184 | 0.1917 | 0.2021 | 0.2450 | 0.2541 |
Table 7 provides estimates of parametric coefficients with respect to the two models. For both models, the coefficients are statistically significant.
Model | Coefficient | Standard Error | Coefficient/Standard Error | Probability of a Type 1 Error |
Model 1 | 11.85 | 1.34 | 8.841 | $\approx 0$ |
Model 2 | 11.48 | 0.2506 | 45.8 | $\approx 0$ |
In the case of a linear regression model, the goodness of fit is indicated by the residual sum of squares. However, in the case of GAMs, the sum of squares does not serve as a credible measure of the difference between the observed values of the response variable and the fitted values. In lieu of this, the discrepancy in terms of deviance can be measured.
$D=2\left[L\left(\hat{\beta}_M\right)-L\left(\hat{\beta}_O\right)\right] \varphi$
where, $L\left(\hat{\beta}_M\right)$ is the maximum likelihood of the saturated model, $L\left(\hat{\beta}_O\right)$ is the maximum likelihood of the observed (fitted) model, and $\Phi$ is the scale parameter.
In a statistical analysis, a model is validated through cross-validation (CV) i.e. testing on the basis of data not used previously. The principal purpose of CV is to judge the ability of the selected model to correctly predict new response data points which was not utilised in the regression process. The CV procedure incorporates segmenting the available data into two complementary (mutually exclusive) subsets: in-sample which is used for carrying out the regression analysis and the second which is utilised for testing the model. Repeated CV is normally done in order to decrease heterogeneity in test results. In the present study, the GCV (Generalised Cross-Validation) score of the fitted GAM is computed. Lower values of GCV imply better reproducibility of the data.
Table 8 exhibits the results of the two models used in the present study. A comparison of the results indicates that Model 1 exhibits slightly better goodness of fit compared to Model 2.
Particulars | Model 1 | Model 2 |
Adjusted | 0.95 | 0.939 |
Deviance explained | 95.5% | 94.5% |
GCV | 0.0538 | 0.0672 |
Scale estimate | 0.0484 | 0.0598 |
Table 9 presents the variance-component estimates for the two models. The table presents the overall standard deviation $\sigma=\sqrt{\sigma_u^2+\sigma_v^2}$ and the ratio of the SD (std. deviation) of asymmetric(one-sided) and symmetric disturbance terms $\lambda=\frac{\sigma_u}{\sigma_v}$. The results imply that the total standard deviation is slightly lower in the first model compared to the second one. Further, the ratio of the standard deviation due to internal factors and external factors is higher in Model 1 relative to Model 2.
Particulars | Model | Estimate |
Lambda $(\lambda)$ | Model 1 | 2.3543 |
Sigma $(\sigma)$ | 0.3077 | |
Lambda $(\lambda)$ | Model 2 | 1.7865 |
Sigma $(\sigma)$ | 0.3214 |
A part of this research enquiry is to explore the relationship of profitability measures (ROE and ROA) with revenue efficiency. However, owing to the existence of omitted variables, the application of ordinary least squares cannot be fruitful as the efficiency variable is correlated with the error term of the model. Therefore, the instrumental variable method of regression is applied. The solvency ratio is used as the instrumental variable, which has a very low correlation with the dependent variables (ROE and ROA) but is highly correlated with the explanatory variable. Table 10 through Table 13 present the instrument variable regression outcomes for the two models. To be more specific, Table 10 presents the results with respect to model 1 with ROE as the dependent variable. Table 11 presents the results for model 2 with ROA as the dependent variable. Table 12 and Table 13 represent the results for model 2 with the dependent variables being ROE and ROA, respectively.
Particulars | Coefficient | Coefficient Lower Bound | Coefficient Upper Bound | Standard Error | t-ratio | Probability of a Type 1 Error |
Constant | -0.3869 | -0.8208 | 0.4096 | 0.2214 | -1.748 | 0.0840 |
Efficiency | 0.9750 | 0.0469 | 1.5405 | 0.2885 | 3.380 | 0.0011 |
Particulars | Coefficient | Coefficient Lower Bound | Coefficient Upper Bound | Standard Error | t-ratio | Probability of a Type 1 Error |
Constant | -0.0262 | -0.0939 | 0.0415 | 0.0346 | -0.7590 | 0.4499 |
Efficiency | 0.1403 | 0.0542 | 0.2263 | 0.0439 | 3.193 | 0.0020 |
Particulars | Coefficient | Coefficient Lower Bound | Coefficient Upper Bound | Standard Error | t-ratio | Probability of a Type 1 Error |
Constant | -0.4165 | -0.7727 | -0.0603 | 0.1817 | -2.292 | 0.0243 |
Efficiency | 1.0127 | 0.5268 | 1.4986 | 0.2479 | 4.085 | <0.0001 |
Particulars | Coefficient | Coefficient Lower Bound | Coefficient Upper Bound | Standard Error | t-ratio | Probability of a Type 1 Error |
Constant | -0.0305 | -0.0756 | 0.0147 | 0.0230 | -1.324 | 0.1891 |
Efficiency | 0.1457 | 0.0868 | 0.2046 | 0.0301 | 4.847 | <0.0001 |
The data used to support the findings of this study are available from the corresponding author upon request.
The author declares that there is no conflict of interest.
Table A1. Revenue efficiency scores for Model 1
Insurer | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Bajaj Allianz | 0.7363 | 0.7883 | 0.8517 | 0.9008 | 0.8630 | 0.8162 |
Cholamandalam | 0.9210 | 0.8952 | 0.8452 | 0.8035 | 0.6841 | 0.7559 |
Future Generali | 0.8892 | 0.8992 | 0.9020 | 0.8486 | 0.7862 | 0.7002 |
HDFC Ergo | 0.7690 | 0.7332 | 0.7947 | 0.7086 | 0.5883 | 0.6212 |
ICICI Lombard | 0.8736 | 0.8759 | 0.7992 | 0.8242 | 0.6635 | 0.7293 |
IFFCO Tokio | 0.8573 | 0.8218 | 0.8675 | 0.9344 | 0.9074 | 0.9243 |
Reliance | 0.7168 | 0.7611 | 0.8344 | 0.8540 | 0.8008 | 0.7472 |
Royal Sundaram | 0.8953 | 0.8485 | 0.9098 | 0.8863 | 0.8568 | 0.8606 |
SBI General | 0.3017 | 0.5881 | 0.7522 | 0.7793 | 0.6174 | 0.7677 |
Shri Ram General | 0.9331 | 0.7267 | 0.9404 | 0.9117 | 0.9068 | 0.9170 |
Tata AIG | 0.8722 | 0.8654 | 0.8864 | 0.9268 | 0.8592 | 0.8064 |
National | 0.8498 | 0.6692 | 0.8275 | 0.7634 | 0.8025 | 0.8032 |
New India | 0.8055 | 0.9612 | 0.8453 | 0.8639 | 0.8821 | 0.9236 |
Oriental | 0.8398 | 0.7504 | 0.7952 | 0.6909 | 0.7263 | 0.7795 |
United | 0.8441 | 0.7648 | 0.7547 | 0.7199 | 0.7787 | 0.8365 |
Source: Calculated by the author.
Table A2. Revenue efficiency scores for Model 2
Insurer | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Bajaj Allianz | 0.9160 | 0.9125 | 0.8754 | 0.8608 | 0.8175 | 0.8213 |
Cholamandalam | 0.9450 | 0.9148 | 0.8534 | 0.8249 | 0.7877 | 0.8798 |
Future Generali | 0.9390 | 0.9244 | 0.9113 | 0.8384 | 0.7799 | 0.6880 |
HDFC Ergo | 0.7983 | 0.7430 | 0.8078 | 0.8180 | 0.7751 | 0.7981 |
ICICI Lombard | 0.9063 | 0.8720 | 0.8041 | 0.8136 | 0.7010 | 0.6948 |
IFFCO Tokio | 0.8257 | 0.8160 | 0.8679 | 0.9353 | 0.9230 | 0.8791 |
Reliance | 0.6986 | 0.7414 | 0.8316 | 0.8395 | 0.7856 | 0.6286 |
Royal Sundaram | 0.8736 | 0.8096 | 0.8432 | 0.8086 | 0.7683 | 0.8190 |
SBI General | 0.3796 | 0.6810 | 0.7224 | 0.6881 | 0.5913 | 0.7396 |
Shri Ram General | 0.9028 | 0.6100 | 0.7900 | 0.6998 | 0.6088 | 0.5613 |
Tata AIG | 0.8510 | 0.8251 | 0.8427 | 0.8948 | 0.8815 | 0.8980 |
National | 0.8637 | 0.7577 | 0.8344 | 0.8736 | 0.9000 | 0.8608 |
New India | 0.8161 | 0.9179 | 0.8225 | 0.8165 | 0.8343 | 0.8150 |
Oriental | 0.8611 | 0.8429 | 0.8189 | 0.7934 | 0.8498 | 0.8476 |
United | 0.7379 | 0.7679 | 0.7751 | 0.7865 | 0.8243 | 0.8003 |
Source: Calculated by the author.
Table A3. Technical (revenue) inefficiency scores for Model 1
Insurer | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Bajaj Allianz | 0.3061 | 0.2379 | 0.1605 | 0.1045 | 0.1473 | 0.2031 |
Cholamandalam | 0.0823 | 0.1107 | 0.1682 | 0.2188 | 0.3797 | 0.2799 |
Future Generali | 0.1174 | 0.1062 | 0.1031 | 0.1642 | 0.2406 | 0.3564 |
HDFC Ergo | 0.2626 | 0.3104 | 0.2298 | 0.3444 | 0.5305 | 0.4761 |
ICICI Lombard | 0.1351 | 0.1325 | 0.2241 | 0.1934 | 0.4102 | 0.3157 |
IFFCO Tokio | 0.1540 | 0.1962 | 0.1421 | 0.0678 | 0.0972 | 0.0787 |
Reliance | 0.3329 | 0.2730 | 0.1810 | 0.1578 | 0.2221 | 0.2914 |
Royal Sundaram | 0.1106 | 0.1643 | 0.0945 | 0.1207 | 0.1546 | 0.1501 |
SBI General | 1.1982 | 0.5308 | 0.2848 | 0.2493 | 0.4823 | 0.2643 |
Shri Ram General | 0.0692 | 0.3193 | 0.0615 | 0.0924 | 0.0978 | 0.0866 |
Tata AIG | 0.1367 | 0.1446 | 0.1206 | 0.0760 | 0.1518 | 0.2152 |
National | 0.1627 | 0.4017 | 0.1894 | 0.2700 | 0.2200 | 0.2192 |
New India | 0.2163 | 0.0396 | 0.1681 | 0.1463 | 0.1255 | 0.0795 |
Oriental | 0.1746 | 0.2871 | 0.2291 | 0.3698 | 0.3198 | 0.2491 |
United | 0.1695 | 0.2681 | 0.2814 | 0.3286 | 0.2501 | 0.1785 |
Source: Calculated by the author.
Table A4. Technical (revenue) inefficiency scores for Model 2
Insurer | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
Bajaj Allianz | 0.9160 | 0.9125 | 0.8754 | 0.8608 | 0.8175 | 0.8213 |
Cholamandalam | 0.9450 | 0.9148 | 0.8534 | 0.8249 | 0.7877 | 0.8798 |
Future Generali | 0.9390 | 0.9244 | 0.9113 | 0.8384 | 0.7799 | 0.6880 |
HDFC Ergo | 0.7983 | 0.7430 | 0.8078 | 0.8180 | 0.7751 | 0.7981 |
ICICI Lombard | 0.9063 | 0.8720 | 0.8041 | 0.8136 | 0.7010 | 0.6948 |
IFFCO Tokio | 0.8257 | 0.8160 | 0.8679 | 0.9353 | 0.9230 | 0.8791 |
Reliance | 0.6986 | 0.7414 | 0.8316 | 0.8395 | 0.7856 | 0.6286 |
Royal Sundaram | 0.8736 | 0.8096 | 0.8432 | 0.8086 | 0.7683 | 0.8190 |
SBI General | 0.3796 | 0.6810 | 0.7224 | 0.6881 | 0.5913 | 0.7396 |
Shri Ram General | 0.9028 | 0.6100 | 0.7900 | 0.6998 | 0.6088 | 0.5613 |
Tata AIG | 0.8510 | 0.8251 | 0.8427 | 0.8948 | 0.8815 | 0.8980 |
National | 0.8637 | 0.7577 | 0.8344 | 0.8736 | 0.9000 | 0.8608 |
New India | 0.8161 | 0.9179 | 0.8225 | 0.8165 | 0.8343 | 0.8150 |
Oriental | 0.8611 | 0.8429 | 0.8189 | 0.7934 | 0.8498 | 0.8476 |
United | 0.7379 | 0.7679 | 0.7751 | 0.7865 | 0.8243 | 0.8003 |
Source: Calculated by the author.