A survey of life insurance efficiency papers: Methods, pros & cons, trends

Article history: Received October 5, 2016 Received in revised format November 11 2016 Accepted November 2


Introduction
This paper explores the methods most commonly used in over 190 studies that determine life insurance efficiency, an area that is gaining in recognition as being important to investigate.In addition an evaluation of the different approaches is undertaken as well as an examination of the numbers and recent trends of methods and some aspects of life insurance efficiency measurement.This article enhances and improves upon the more limited studies such as Berger and Humphrey (1997), Cummins and Weiss (2000) and Eling and Luhnen (2010a) making contributions such as an overview of life insurance efficiency studies performed since 1983 and guidance as to the advantages and disadvantages of the different techniques used therein plus their applicability to life insurance.This paper also shows how the most fundamental elements of life insurance efficiency estimation, such as the set-up and form of outputs and inputs, have been coped with.Therefore an assessment of the overall results of efficiency studies is possible.Additionally ideas for potential further research are discussed.Consequently this review will be useful to both practitioners and regulators concerned with this area.This article continues with Section 2 which describes the most common methods of determining efficiency and some of their main advantages and disadvantages.Section 3 discusses output and input proxies used in life insurance efficiency studies while Section 4 exhibits the numbers and trends of papers, output proxies and input proxies.Section 5 concludes and postulates ideas for possible further research.

Most Common Methods of Determining Efficiency
The most common methods utilized to determine life insurer efficiency number seven comprising two nonparametric, three parametric, one semi-parametric and the Bayesian.As true efficiency is unknown it is impossible to tell which approach gives best outcome (Hussels & Ward, 2004) so the process chosen should depend on items such as the purpose of the study and the type of data available (Hjalmarsson et al., 1996).
The two nonparametric approaches, data envelopment analysis (DEA) and free disposal hull (FDH), do not specify a form for the underlying production relationship between inputs and outputs.The linear programming technique DEA creates frontier observations with no other (linear combination) of decision making units (DMUs) having at least as much (little) output (input) for a given set of inputs (outputs).The technique is called data envelopment because the data of the most efficient DMU "envelops" the data of the others.FDH, introduced by Deprins et al. (1984), is a special case of DEA in that the latter allows for linear combinations of observed input sets and FDH assumes no such replacement is possible (Bauer et al., 1998;Saad et al., 2006).Hence the production possibilities of FDH are only the vertices calculated incorporating DEA and the points calculated using FDH that are interior to them.This leads to the average efficiency estimated by the FDH method being at least as large, and often larger, than that when applying DEA.
The three parametric techniques, stochastic frontier analysis (SFA), thick frontier analysis (TFA) and the distribution-free approach (DFA), specify a functional form for the efficiency frontier.The DFA evaluates each firm's inefficiency as the difference between its average residual and that of an institution on the efficiency frontier.This results in the DFA assuming that over time the efficiency of each company exhibits little change and the "random errors average to zero" (Berger & Humphrey, 1997;Schmidt & Sickles, 1984).
SFA sets each firm's inefficiency as its residual from the efficiency frontier and usually takes the form Mi = f(ki; β) exp(vi -ui), (1) where f(ki; β) is the functional form of the efficient frontier, Mi is the measured value, the ki values are independent variables and the β parameters are to be estimated.Finally, noise is represented by exp i v and exp i u represents inefficiency.These latter two form the residual, which therefore has two pieces, a random error term and the efficiency term with the former usually assumed to follow a symmetric distribution, such as the standard normal.The efficiency term is usually set as a non-negative asymmetric distribution such as the half-normal, truncated normal or gamma.
TFA is similar to SFA except that it assumes that deviations from the predicted efficiency within the highest and lowest quartiles, quintiles or other sets of the observations that are utilized represent random error.The deviations between these sets may occur in either the intercepts or in the slope parameters (Bauer et al., 1998) and thus represent inefficiencies.Therefore the frontier is thick in the sense that on a graph it would not be seen as a line but as a polygon.As opposed to point calculations for individual firms TFA gives an assessment of the general level of the overall efficiency of an industry.
Fully parametric approaches employ a fully specified model with the full distributions of vi and ui being known up to the specific values of the parameters.In contrast the semi-parametric methods keep the essential structure of the stochastic frontier but relax either one assumption restriction in the model or a specific distribution for vi and/or ui (Greene, 2008).To keep the structure of the stochastic frontier a functional form is needed with the most common used being the Fourier flexible (FF).Here the SFA functional form Mi = f(ki; β) is a "kernel" with terms incorporating sine and cosine functions added and because the sine and cosine functions are orthogonal, the FF functional form can globally approximate any function well.
The Bayesian procedure utilizes information, e.g. from economic theory, to estimate the model parameters.The estimation is called a prior probability distribution function (pdf).With SFA, for example, the prior distribution may be p (β,σ) where β represents parameters β1, β2, β3,….The likelihood function L(x │β,σ) (x represents the data points x1, x2 x3,…) is next calculated.Then as Bayes' Theorem shows that the posterior pdf p(β,σ│x) α L(x │β,σ) p(β,σ), a marginal pdf p(βk│x) for each element of β can usually be determined and the probability that βk lies in an interval evaluated.

The Nonparametric Methods -Advantages and Disadvantages
One advantage of the nonparametric methods is that they are simple and associated with easy calculations (Coelli et al., 2005) as they do not necessitate the full specification of the functional form or distributions of the inefficiency or random noise terms (Berger & Humphrey, 1997;Cummins et al., 2010;Leverty et al., 2004).As a result specification errors (Cummins et al., 2010) and many subjective features are circumvented (Qiu & Chen, 2006).In addition Sinha and Chatterjee (2009) and Trigo Gamarra & Growitsch (2008) note that not requiring a specific functional form of the underlying technology makes the nonparametric approaches especially useful when looking at service industries such a life insurance because for them there is limited information about said underlying production technology (Fecher et al., 1993).
An advantage of DEA specifically is that it can be applied to distinguish between (pure) technical efficiency versus scale efficiency and allocative efficiency (Chen et al., 2008;Cummins et al., 1999a;Hardwick et al., 2003).It can also allow for the assessment of the directions and potential for improvement for each inefficient DMU (Chen et al., 2008;Cummins & Weiss, 2000).Other advantages of DEA are that it results in consistent estimators (Banker, 1993) even when the variances of regression disturbances exhibit heteroscedasticity (Banker et al., 2004).Furthermore the asymptotic distribution of the DEA inefficiency estimators is the same as their true distribution (Banker, 1993).
A second nonparametric method, FDH, was introduced by Deprins et al. (1984), as they disapproved of DEA assuming a convexity of the efficient frontier.The FDH technique eliminates the convexity assumption (Naini & Nouralizadeh, 2012) even though some industries oblige it (Cummins & Weiss, 2000).Another advantage of FDH is that its less arbitrary assumptions may lead to a more accurate fit to the data than does DEA (Tulkens, 1993;Vanden Eeckaut, Tulkens & Jamar, 1993).Consequently efficiencies calculated employing the FDH approach are actual observations as opposed to DEA's incorporation of a built production frontier and so seem more credible (Tulkens, 1993).Moreover FDH removes the bias of DEA due to fully efficient DMUs being utilized repeatedly in DEA to create more theoretical firms (Ennsfellner et al., 2004).
The main drawbacks with nonparametric methods are the use of a deterministic procedure and an assumption of no random error.Berger & Humphrey (1991) notes that efficiency frontier construction incorporates an assumption of no measurement errors and, more importantly, a change in the measured efficiency of a DMU not depending on good or bad luck (Berger & Mester, 1997;Cummins & Weiss, 2000;Simar & Wilson, 1998).With respect to life insurance as almost every facet of it, e.g.mortality, morbidity, lapse and interest rates, contains a large element of randomness, excluding randomness presents problems.Furthermore if some companies get very lucky results (i.e.not due to capability) then the efficiency scores of other companies will be unduly low (Bauer et al., 1998).One more difficulty is that these techniques ignore accounting rules which distort the appraisal of inputs or output values (Berger & Humphrey, 1997).The outcome of the foregoing problems is that any calculation anomalies are recorded as a change in efficiency rather than a computation error.Additionally efficiency estimation of other DMUs may be distorted if compared to one suffering from one of these problems (Berger & Humphrey, 1991).DEA assumes that the range of available inputs is similar across all DMUs (Dyson et al., 2001).However this is not true in many life insurance markets.For example in the United States life insurance industry there is a large disparity in company asset size as shown by the largest firm in 2014 having over $390 billion in assets and the smallest having less than $100 thousand.The huge difference is further demonstrated as the tenth largest life insurer by asset size has over $161 billion and the 100 th has just over $10 billion in assets.In Canada the largest company in 2014 had over $190 billion in assets and the smallest had less than $3 million and for Australia the corresponding values were $87 billion and $13 million.Hence, for example, the types of assets obtainable to smaller firms may not be the same as for the larger.1 Also, institutions issuing different lines of business or that are in different locations may not draw upon or have available the same inputs (Barros et al., 2005a).
Another significant drawback with DEA and FDH is that they were designed to be applied to not-forprofit DMUs (Charnes et al., 1978) that do not have the usual economic goals such as profit maximization or cost minimization (Sun & Zhong, 2011).Additional deficiencies include that 1) the frontier can be shaped by the data,2 2) the calculation is very susceptible to the number of exogenous constraints used (Berger & Humphrey, 1991;De Luca Cardillo & Fortuna, 2000) and to input/output specification and outliers (Ennsfellner et al., 2004;Deng, 2010), 3) there is no accounting for input or output prices and so no evaluation of allocative inefficiency (Berger & Mester, 1997) thus leading to an upward bias in efficiency scores (Berger & Humphrey, 1997;Simar & Wilson, 1998), 4) because, as Cummins & Weiss (2000) remarks, relative prices cannot be used to compare non-alike companies as only the data of entities closest in type to that being assessed are used in quantifying the inefficiency of said entity (Avkiran, 2002;Berger & Mester, 1997), 5) firms can have very high efficiency scores simply because few others have analogous inputs, outputs or related observations (Bauer et al., 1998), 6) the performance of the interaction between components of a DMU cannot be determined (Kao, 2009;Sexton & Lewis, 2003), 7) an underlying model or reference technology results in a bias in DEA assessments (Kittelsen, 1999) and 8) different returns to scale assumptions in any underlying technology lead to completely different conclusions (Fare et al., 1994;Ray & Desli, 1997).The outcome of some of these problems is that nonparametric methods allow only for an analysis of technological, not economic, optimization (Berger & Humphrey, 1997;Huang, 2007).

Overcoming the Disadvantages of the Nonparametric Methods
Several processes of overcoming the aforementioned drawbacks of DEA are employed in the life insurance efficiency literature.Some of the most common of these are the utilization of bootstrapping, slacks, range adjustment and an assurance region.
Bootstrapping is designed to overcome the problem of uncertainty related to the measurements arrived at via the nonparametric approaches (Mahlberg & Url, 2010;Ubl, 2010), which was seemingly not considered (Assaf et al., 2012).The bootstrap procedure was introduced by Efron (1979) as more applicable and dependable than the "jackknife."The idea is to investigate the sensitivity of efficiency estimates to differences in data samples by first replicating the data generating process (DGP).Then the original estimator is applied to each replication.In this way the simulations can be applied to imitate the original (unknown) sampling distribution (Berger & Humphrey, 1997, Simar & Wilson, 1998).The bootstrap of the DGP approximately replicates the variation due to sampling of the calculated efficiency frontier, therefore permitting an exploration of its sensitivity (Simar & Wilson, 1998) and associated confidence intervals (Simar & Wilson, 2000).Bootstrapping enables the testing of concepts such as statistical significance of the disparities in efficiency estimates, statistical inference and bias correction of estimators (Simar & Wilson, 2000;Leverty et al., 2009) and hypotheses regarding the underlying technology (Badunenko et al., 2006;Simar & Wilson, 1998).
Even so bootstrapping has the disadvantage that it accentuates the problem that the nonparametric methods do not account for random noise of the data (Simar & Wilson, 1998) in that addition more noise is introduced into the data with bootstrapping (Simar & Wilson, 2008).DEA incorporates a radial technique which has a fundamental drawback in that it does not illustrate all of the input decreases or output increases, i.e. "slacks" (Sinha, 2007a).Hence it cannot identify potential efficiency increases from such changes (Tone, 1998).Tone (1998) introduced a slacks-based DEA method to overtly include input overindulgences and output deficiencies in the objective function (Drake et al., 2006).The consequence is that if a DMU has larger slacks than a second, the first is deemed less efficient (Tone, 1998) such that with slacks inefficiency can be measured as a product of input and output inefficiencies (Sinha, 2015).So the optimal result is when a DMU has no input and output slacks (Drake et al., 2006).
The range-adjusted DEA approach builds on slacks based DEA.The inefficiency scores calculated are "dimensionless" with the first step being dividing each slack variable by its range over the DMUs.Then, using the set-up leading to the initial DEA efficiency scores, the sum of 1) the above quotients over 2) the total number of inputs and outputs ε [0,1] which becomes the inefficiency scores of the DMUs.Accordingly a DMU is deemed to be fully efficient if and only if there are no slacks in any inputs or outputs (Brockett et al., 2005;Leverty & Grace, 2008).Positive characteristics of the rangeadjusted method include that the efficiency scores do not alter if either the location or scale of outputs or inputs change (Brockett et al., 2005) and said scores are robustly monotonic and can thusly be used to rank DMUs (Cooper et al., 1999;Leverty & Grace, 2008).
Another mechanism of dealing with slacks is assurance region DEA, introduced by Thompson et al. (1986), which limits input and output shadow prices.The limits are achieved by setting bounds on the ratios of input and output shadow prices to each other (Sinha, 2007a).Therefore the isoquant is changed so that, at the most efficient points, slacks cannot be present on the radial projection of any combination of inputs and outputs onto the changed isoquant.If such slacks are present there is too much of either an input or output which puts the linear programming solution, found in terms of the ratios of the shadow prices, outside the set shadow price restriction (Sinha, 2007a;Thompson et al., 1986).

The Parametric Methods -Advantages and Disadvantages
Much like the nonparametric, the parametric methods have both advantages and disadvantages.Other than advantages corresponding to the foregoing disadvantages of the nonparametric methods, an advantage of the parametric techniques is that they absorb some effect of heterogeneity in inputs and outputs (Cummins & Weiss, 2000).Other advantages include that they enable statistical testing of hypotheses and calculating confidence intervals (Hjalmarsson et al., 1996).
The main difficulty with the parametric approaches is the necessity of correct functional form and error term distributions to obtain unbiased parameter estimates.Not assuming the correct form or distributions can lead to specification errors such that 1) either efficiency determinations can be mixed up with said specification error (Bauer et al., 1998;Berger & Humphrey, 1991;Cummins & Weiss, 2000) or 2) either the efficiency or random error measures do not fit the observed data (Berger & Humphrey, 1992;Greene, 1990;Stevenson, 1980).Moreover the parametric methods require the identification of a production, cost or profit function where the largest problem is separating the efficiency scores from luck and random error properly (Berger & Humphrey, 1992).As well, such a function assumes an underlying production relationship, which may not be true (Drake et al., 2006;Hjalmarsson et al., 1996).Some advantages and disadvantages of specific parametric techniques are described in the following.
SFA has key advantages in that it can differentiate between efficiencies and measurement error (Koop et al., 1994) and exhibits internal consistency and furthermore is easy to apply (Greene, 2008).But notwithstanding these any difficulty concerning the specification errors are emphasized with the utilization of the two error terms which must be separated properly (Koop et al., 1994).
DFA was originated in Schmidt & Sickles (1984) and Berger (1993) with one advantage being that it only involves a small amount of theoretical assumptions with respect to the data and the production process (Ryan, Jr., & Schellhorn, 2000).One more plus concerns how DFA handles random error.DFA does not assume a distribution of random error (as does SFA) and DFA does not assume that differences between groups of companies are all inefficiencies (as does TFA) (Bauer et al., 1998).
However DFA has the same drawback regarding random error as does DEA.Another problem with DFA, originally pointed out in Schmidt & Sickles (1984), is that as average residuals are employed a change in, for example, technology or regulations affecting the efficiency of all DMUs examined results in the DFA estimating each company's average inefficiency over time (Berger et al., 1997;Berger & Humphrey, 1997).Such an evaluation is problematic as it is more desirable to appraise efficiency against the frontier at one point in time such as just before or after said change.
TFA has an advantage in that it needs little in the way of assumptions and so, compared to SFA, may be less prone to the specification errors mentioned previously (Berger & Humphrey, 1992;Ennsfellner et al., 2004).For instance there is no requirement for the regressors to be uncorrelated with the efficiencies (Berger & Humphrey, 1992) and the only assumptions necessary as to efficiency and random error are that the highest and lowest quartiles incorporate different efficiencies and that there is random error within said quartiles (Berger & Humphrey, 1997).In addition there is less likelihood of the bias seen in DEA efficiency estimates (Berger & Humphrey, 1991) and TFA is not subjected to the influence of outliers (Bikker & van Leuvensteijn, 2008).
A weakness of TFA is that the sets used may be determined using the dependent variables of the regressions which can bias the coefficient estimates (Berger & Humphrey, 1992).As well TFA requires data that is highly dispersed (Ennsfellner et al., 2004).

The Semi-Parametric and Bayesian Methods
Semi-parametric methods have an advantage in that the properties of the cost or profit function can be established from the data (Koop et al., 1994).A deficiency of the semi-parametric methods is that efficiency calculations can be very misleading if an inappropriate functional form is chosen.When using ui and vi terms their separation is an important consideration and can be the least robust to arbitrary assumptions (Koop et al., 1994).
As stated earlier the most common functional form applied with the semi-parametric method is the FF.One disadvantage of utilizing a FF functional form is that the sine and cosine functions of the FF form have no economic interpretation making it difficult to analyze any outcomes obtained.Moreover the sine and cosine functions do not satisfy the usual regularity conditions, such as increasing monotonically and being strictly quasi-concave (Yue, 1991), even though this drawback can be overcome by employing the procedure of Gallant & Golub (1982), forcing quasi-convexity of the consumer's individual utility function to easily make the FF functional form regular (Barnett et al., 1991).Furthermore, a FF form can overfit the random error contained in the data (Koop et al., 1994;Yue, 1991) as a large enough FF functional form will ultimately attain a perfect fit because noise will be looked at as irrational behavior (Barnett et al., 1991).Also because n-order trigonometric terms3 are included there is an increased chance of multicollinearity among the function's terms which hinders an assessment of the meaning of the coefficient estimates (Ward, 2002).Additionally Altunbas & Chakravarty (2001) reports that, even though compared to a translog functional form, a FF functional form may have a better fit to the data; it may have a worse predictive ability.In fact Marie et al. (2009) finds that translog form outperformed the FF form.
There are other semi-parametric methods in the literature.One of these is the Muntz-Szatz expansion of Barnett et al. (1991) that Koop et al. (1994) relates fits only that part of the data that is globally regular, thereby eliminating the risk of overfitting.
Another semi-parametric process, used by Fan et al. (1996), is based on a production model where g(xi; β) is the functional form of the efficient frontier, yi represents the outputs, and xi the inputs of firm i and the β parameters are to be estimated.Finally, noise is represented by exp i v and exp i u represents inefficiency.In the Fan et al. (1996) model the functional form is g(xi; β) = w′i β + m(zi) where the functional form of m(.) is unknown.
A third semi-parametric technique, incorporated by Park & Simar (1992), is where yi represents the outputs, and xi the inputs of firm i with the β parameters to be estimated.B(h) is the upper bound of the unknown density h.The αi -B(h) corresponds to the technical inefficiency of the firm i (with the αi being iid from h) and εit is noise.The paper gives an asymptotic lower bound of β and an efficient estimator of β that attains said lower bound.Then the predictors of αi are built using the β estimates.Lastly an estimator of B(h) is shown which gives estimates of the frontier function and so the technical inefficiencies.Adams et al. (1999) specifies a semi-parametric approach similar to Park & Simar (1992).This study begins with the panel-data model where yi represents the outputs, y*i represents the normalized (by the last yi) outputs, and xi the inputs of firm i and the β and γ parameters are to be estimated.Finally αi represents the constant level of inefficiency and εit is noise.Adams et al. (1999) draws upon a semi-parametric method where no parametric assumptions are made for the inputs.This procedure allows the forcing of necessary restrictions, particularly having the output distance function be linear homogenous, on outputs and so lets a correlation between a subset of the regressors and efficiency scores be set up (Adams et al., 1999).
One difficulty with the Bayesian approach is the need to choose a reasonable prior pdf without which the estimates with respect to each βk may be useless or nonsensical.Moreover the prior pdf is selected by the researcher which can lead to problems such as bias or error in their views.As well it may be difficult to calculate the marginal pdfs as doing so can require complex integration.

Output and Input Proxies
As efficiency is an evaluation of the ability of a company to manufacture outputs from inputs it is necessary to designate measures to use as output and input proxies (Ennsfellner et al., 2004;Leverty et al., 2009).The difficulty regarding life insurance is, as its output is intangible services, the output volume must be approximated by proxy variables (Leverty et al., 2004;Weiss, 1986).However there is a debate in the literature as to which of the two basic sets of prevalent output proxies used, 1) reserves (or their change) and claims and 2) premiums and investment income is more appropriate.
Reasons given for utilizing (change in) reserves include that 1) such a value is the best proxy for the volume of underwriting, claims handling and other real services as it is highly correlated with both the numbers of claims and policies (Cummins et al., 1999a;Klumpes, 2006;Leverty et al., 2004), 2) reserves accounts, as a supplement to past losses accounted for by using claims, for expected future losses (Cummins & Rubio-Misas, 2001;Kim & Grace, 1995) and 3) the change in reserves is good proxy for the intermediation of the concurrent year because of the idea that the reserve value will equal the value of assets held by the company (Cummins et al., 1999a;Karim & Jhantasana, 2005;Trigo Gamarra & Growitsch, 2008).
Claims is linked with the use of (change in) reserves as an output proxy4 with the rationale including 1) claims represent payments received by policyholders and are good proxies as they measure the amount of funds pooled and redistributed (i.e. for losses) by insurers (Berger et al., 2000;Cummins et al., 1999a;Tone & Sahoo, 2005), 2) that such redistribution is the object of risk-pooling (Cummins & Rubio-Misas, 2001;Tone & Sahoo, 2005), 3) versus reserves representing future expected losses, claims equal current expenses and losses (Cummins et al., 2004;Trigo Gamarra, 2008) and 4) claims are a good proxy for real services as the amount of claims settlement and real management services are highly correlated with loss amounts (Berger et al., 2000;Cummins et al., 1999a).
The other basic set of output proxies is premiums and investment income.Considering life insurance Blair et al. (1975, p. 185) says that "[p]remiums written has been selected as the measure of output size, which is analogous to using dollar sales volume as a proxy for output" and Fecher et al (1993, p. 81) states that "[p]remiums collected directly concern the technical activity of an insurance company.It reflects the ability of an insurance company to market products, to select clients, and to accept carrying risks."Other life insurance research making similar statements regarding premiums as output include Hussels & Ward (2004) and Ward (2002).Furthermore Diewert (1995, p. 41) explains that "gross premiums paid rather than net [i.e. of claims] premiums … is in agreement with our suggested nominal measure of output" and Hu et al. (2009) points out that premiums are the basis for insurer expenses and profits.
With respect to annuities Segal (2002, p. 84) remarks that "[a]ssuming a positive spread, the larger the annuity considerations, the higher is the expected profit.Hence, a plausible proxy for this output is annuity considerations, which represent the increase in the earnings base of this line of business."As accident and sickness mostly takes into account risk (as opposed to intermediation) if "the risk associated with such policies is priced correctly, premiums are a good proxy for risk" (Segal, 2002, p. 84).Some papers including Greene & Segal (2004), Mansor & Radam (2000) and Rees et al. (1999) advance alternatives to premiums for an output proxy such as policy count and face value.Reasons given include that 1) premium increases influence the output amounts (Bernstein, 1998), 2) premiums are not quantity of output as they are the product of price and quantity (and so are revenue) (Cummins & Zi, 1997;Leverty et al., 2004;Yuengert, 1993) and 3) there can be premium differences between large and small insurers (Boonyasai et al., 2002;Yuengert, 1993).
Investment income is linked with the use of premiums as a proxy for output.Several studies for instance Berger et al. (2000), Cummins & Rubio-Misas (2001) and Greene & Segal (2004) use asset values as a proxy for output.However investment income is considered by some to be a better proxy because it is a flow value rather than a static value.In addition investment income gives an idea of the expertise of insurers concerning their investment competence (Wu & Zeng, 2011).Hussels & Ward (2004, p. 9) agrees as "life insurance companies collect funds in advance of paying benefits [and t]he process of working with the[se] funds during the time lag is referred to as the intermediation service." The treatment of inputs is less varied than outputs as labor and capital is recognized by virtually all writers.The other values incorporated as input proxies vary somewhat with material and/or business services, or similar terminology, being most common.

Number of Papers
The number of papers that have explored life insurance efficiency has steadily increased as seen in Fig. 1 below which shows the number of studies in the survey year-by-year (starting in 1992): One paper from each of 1983 and 1986 and seven from 2015 also in the survey The fact that, regarding life insurer efficiency, there has been a greater diversity of nations analyzed seems to indicate a greater awareness of its significance.

Methods of Determining Efficiency
Of the seven techniques listed above, when disallowing duplication of authors DEA is used in the greatest number of different papers, eighty-four.SFA is utilized in the next highest number, thirty-five different studies.Still disallowing author repetition two articles that incorporate SFA also employ FF as a semi-parametric method; five papers use DFA, two draw on FDH and one utilizes the Bayesian procedure. 5No paper in the survey applies TFA.Eight items incorporate both SFA and DEA while for six studies this concept is not applicable.The counts of methods are shown in Table 1 below: When considering the number of papers both disallowing and with author duplication the trend has been that more have used DEA than SFA in almost all years.The number of studies utilizing the different methods of determining efficiency year-by-year (starting in 1992) are shown in Fig. 6 and Fig. 7 below: Five 2015 papers using DEA and one 2015 paper using SFA also in the survey Of the eight papers that use both SFA and DEA, four illustrate a comparison of the efficiency scores by company characteristics such as company size and organizational form and three only show an overall inefficiency score obtained from the two methods.None of these papers show a company-bycompany comparison of efficiency scores.Now, with respect to employing DEA there is an issue of whether enough DMUs (here insurance companies) are used versus the number of inputs and outputs.If the number of DMUs is low when compared to the number of inputs and outputs a high proportion will have an efficiency score of 100% as they will be difficult to match in all dimensions (Bauer et al., 1998;Thanassoulis et al., 2008).Dyson et al. (2001) notes that if incorporating DEA a study should have a number of DMUs that is at least as large as twice the product of the number of inputs and the number of outputs while Cooper et al. (2001) remarks that the number of DMUs should be at least three times the sum of said two numbers.
For the one hundred and forty-one papers utilizing DEA investigated as to the number of DMUs, inputs and outputs twenty-eight, about twenty percent, do not seem to have enough DMUs to have credibility.
In addition a further nine, about six percent, seem questionable.
Related to the idea of using a parametric method is which functional form should be drawn upon to measure the efficient frontier.the quadratic (Caves & Christensen, 1980).The choice of functional form implies the shape of the isoquants as well as the values of elasticities of factor demand and factor substitution (Greene, 2008).For example, with more than one output a Cobb-Douglas cost function can give rise to all of the possibility frontiers being convex instead of concave which implies only a few outputs being specialized by all DMUs (Greene, 2008).To avoid the above shortcoming flexible functional forms began to be utilized with the translog being the most common, perhaps as not very many confining assumptions concerning the technology are required by it (Kasman & Turgutlu, 2009).Both the translog and the FF functional forms generate a large number of interaction terms and so potentially describe more of the deterministic relationships within the data than does the Cobb-Douglas (Ward, 2002).Even so, these functional forms have flaws such as not being monotonic or globally convex; both of which the Cobb-Douglas cost function exhibits (Greene, 2008) and that the interaction terms can result in a larger possibility of multicollinearity and so a corresponding lower capacity to know what causes costs or profits (Ward, 2002).
Within papers incorporating a parametric method, when disallowing duplication of authors a translog functional form is used in the greatest number of different studies, twenty-nine.Other methods are employed in only fourteen papers.It should be noted that some of the studies draw upon more than one method so the totals of the parametric methods in Table 1 above do not match the total number of papers here.The counts of functional forms are shown in Table 2 below:

Output Proxy Sets
For the efficiency studies of this survey investigated as to the output proxies, when disallowing duplication of authors a set of output proxies (akin to) premiums and investment income is utilized in sixty different studies.Output proxies (akin to) claims and (change in) reserves is employed in fifty different papers.Within the abovementioned sixteen studies draw upon both types of outputs.Four of these display the results when using each type of output proxy separately while the others apply a model that contains (part of) each type.There are seventeen papers for which the concept of output proxy sets is not applicable.The counts of output proxy sets are shown in Table 3 below:  Most papers using an output proxy (akin to) premiums employ only (earned) premiums.Some writers conclude that it is better not to employ such an output proxy but do so for various reasons.There are six such studies.Only a few papers use premiums combined with another measure.The counts of output proxies utilized in papers incorporating those (akin to) premiums are shown in Table 4 below:

Form of Input Proxies
One aspect of the inputs used in life insurance efficiency research is the form of the inputs.The two most prevalent forms are input prices and input monetary values.For the efficiency studies of this survey examined as to input proxies, when disallowing duplication of authors monetary values are used in the greatest number of different studies, fifty-three.Input prices are utilized in forty-two different papers.Fifteen studies incorporate quantities of inputs such as the number of employees, all but one in combination with monetary values.For six of the papers the concept is not applicable.The counts of input proxy forms used are shown in Table 5 below: An important feature of life insurance efficiency studies utilizing input prices is the concept of drawing on such prices that are common (by year) across companies.In papers using input prices, when disallowing duplication of authors, prices that are totally common are incorporated in fifteen different studies.Input prices that are totally not common by year are utilized in fourteen different papers while input prices that are partly common and partly not are used in twenty different papers.The latter are labelled as Mixed in Table 6 below which shows the counts of commonality of input prices: a: For one paper using input prices the commonality is not extant.
When considering the research both without and with author duplication there has no trend as to whether the most studies have used common, not common or mixed input prices.

Conclusions and Further Research
This article has built upon previous studies such as Berger and Humphrey (1997), Cummins and Weiss (2000) and Eling and Luhnen (2010a) to enumerate and investigate the approaches most commonly drawn upon in over 190 papers to determine life insurance efficiency.This article has also described some of the advantages and disadvantages of these methods as well as explored the numbers and recent trends of approaches and some aspects of life insurance efficiency estimation.
There are many possible directions for further research in life insurance efficiency.Perhaps the most important possibility is that of profit efficiency.Only fifteen of the more than one hundred and ninety papers estimate profit efficiency with none written in 2015 or 2014 and only two written in 2013.Only six have been written since 2008.Profit efficiency might be seen as more important than cost efficiency for example Berger et al. (1993) states that profit efficiency may reduce problems associated with misspecification and mismeasurement" and it "allows the researcher to pinpoint better the sources of inefficiency." 6Furthermore profit efficiency is more general than and includes cost efficiency (Akhavein et al., 1997, p. 96) and it may be that that "[i]n studying firm performance, profit maximization is superior to cost minimization" (Berger & Mester, 2003, p. 67).
Another potential for further research is to investigate allocative efficiency in greater detail.Of the seventy-nine papers using input prices only twenty-six show allocative efficiency results, even though other studies state it is possible to calculate the allocative efficiency scores from their cost and technical efficiency scores.Perhaps allocative efficiency scores is being seen as more important as four of the papers that show allocative efficiency scores were written in 2014.
Further research can also be undertaken with respect to the functional forms used to determine life insurer efficiency.For example more of the papers that use both SFA and DEA can display the overall inefficiency scores as well as a company-by-company comparison calculated from the two methods.It also appears clear that more papers should employ both SFA and DEA and compare the results derived as it is possible that they will differ significantly.In addition more methods, such as DFA and FDH, might be used with comparisons of results. 6The emphasis is in the original.
Similar comparisons might be made from the use of different functional forms when using parametric or semi-parametric methods as the results may also vary extensively.Furthermore such comparisons can be performed as to using the two basic sets of prevalent output proxies 1) reserves (or their change) and claims and 2) premiums and investment income.
Another prospect for further research is studies with a focus on developing markets.In addition an analysis of the trends mentioned above may help reveal which are the best directions to take further research.
As life insurance efficiency is an area gaining in recognition as being critical to analyze it can be important to both practitioners and regulators.Therefore those concerned with life insurance efficiency can possibly find this article useful as it will enable an assessment of which aspects of this field of study need more research and which are otherwise worth developing.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .
Fig. 1.Number of Life Insurance Efficiency Studies (Year-by-Year)The steady increase in research investigating life insurance efficiency indicates that it is being thought of as more critical in both the life insurance industry and the financial services industry as a whole.For papers that calculate life insurance efficiency, the focus of papers considering individual nations has changed over the years.When such research began in the early 1990s most examined the efficiency of life insurers in the United States whereas in the late 1990s Germany became a larger focus.Starting in the mid 2000s Asian countries such as the PRC, the ROChina and India along with the United States became the spotlight of more life insurer efficiency studies than other nations.In addition various articles involving multiple nations, such as of Europe, of the Gulf Cooperation Council and worldwide have been performed, especially since the early 2000s.The number of papers of the most focused upon nations year-by-year (starting in 1992) are shown in Fig.2and Fig.3below:

Fig. 6 .Fig. 7 .
Fig. 6.Number of Life Insurance Efficiency Studies Using Different Methods of Determining Efficiency With no Author Duplication (Year-by-Year)

Fig. 9 .
Fig. 9. Number of Life Insurance Efficiency Studies Using Different Output Proxy Sets (Year-by-Year)

Table 2
Counts of functional forms papers using a parametric method Functional Form a: Based on a Translog functional form.
When considering the number of studies without author duplication the recent trend has been that more have utilized a set of output proxies (akin to) premiums and investment income.By a count of papers there is no trend as to which set of output proxies is more prevalent.The number of studies incorporating the different sets of output proxies year-by-year (starting in 1992) are shown in Fig.8and Fig.9below:

Table 4
Counts of output proxies papers using premiums Six papers (five when disallowing author duplication) state that using premiums is inappropriate.

Table 5
Counts of input proxy formsWhen looking at the number of studies both without and with author duplication the trend has been that more have used input monetary values as an input proxy form more recently.The number of papers applying the different input proxy forms year-by-year (starting in 1992) are shown in Fig.10 and Fig. 11 below: One 1983 paper using IP prices and IP monetary values and five 2015 papers using IP monetary values also in the survey Fig. 10.Number of Life Insurance Efficiency Studies Using Different Input Proxy Forms With no Author Duplication (Year-by-Year) One 1983 paper using IP prices and IP monetary values, six 2015 papers using IP monetary values and one paper from each of 1986 and 2015 using IP prices also in the survey Fig. 11.Number of Life Insurance Efficiency Studies Using Different Input Proxy Forms (Year-by-Year)

Table 6
Counts of input price commonality papers using input prices