Naaj 2011 Vol15 No4 Lin Wen Yang

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487

EFFECTS OF RISK MANAGEMENT ON COST

EFFICIENCY AND COST FUNCTION OF THE

U.S. PROPERTY AND LIABILITY INSURERS

Hong-Jen Lin,* Min-Ming Wen, and Charles C. Yang

ABSTRACT

This paper adopts the one-step stochastic frontier approach to investigate the impact of risk man-

agement tools of derivatives and reinsurance on cost efficiency of U.S. property-liability insurance

companies. The stochastic frontier approach considers both the mean and variance of cost effi-

ciency. The sample includes both stock and mutual insurers. Among the findings, the cost function

of the entire sample carries the concavity feature, and insurers tend to use financial derivatives

for firm value creation. The results also show that for the entire sample the use of derivatives

enhances the mean of cost efficiency but accompanied with larger efficiency volatility. Neverthe-

less, the utilization of financial derivatives mitigates efficiency volatility for mutual insurers. This

research provides important insights for the practice of risk management in the property-liability

insurance industry.

1. INTRODUCTION

Insurance companies face both underwriting and investment risks. They tend to utilize appropriatemechanisms to manage the risks. The underwriting risks are attributed to the issuance of insurancepolicies, and insurers commonly apply reinsurance to transfer the risk to reinsurers. On the other hand,

before insurance companies fulfill their obligations to policyholders, premiums collected from policy-holders provide insurers with one source of investment funds. Investment risks are incurred with in-surers engagement in investment activities, and the use of financial derivatives provides insurers a

venue to manage their investment risks.This paper investigates how these two specific risk management tools, financial derivatives and re-

insurance, affect the cost function and cost efficiency of the U.S. property-liability (P/L) insuranceindustry. The estimation of cost function has been an important topic in the study of industrial orga-nization and financial institutions. The functional form of cost function enables us to analyze the degreeof insurers incentives to manage risks. In addition, cost efficiency is an important factor to assess

whether the application of risk management mechanisms can enhance insurers performance. Theenhancement of cost efficiency has become more and more imperative with increasing operationalcosts. It is very interesting and important to examine the effect of risk management on insurers cost

efficiency while insurance companies simultaneously apply financial derivatives and reinsurance to man-age risks.

Corporate risk management helps reduce costs of a company by either reducing the level of costfunction or enhancing cost efficiency. As summarized in Smith and Stulz (1985), the application of

* Hong-Jen Lin is an Assistant Professor in Finance, Department of Finance and Business Management, Brooklyn College, CUNY, Brooklyn, NY,

[email protected]. Min-Ming Wen is an Assistant Professor in Finance, Department of Finance and Law, California State University, Los Angeles, CA,

[email protected]. Charles C. Yang is an Associate Professor in Risk Management and Insurance, Department of Finance, Florida Atlantic University, Boca Raton,

FL, [email protected].


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488 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 15, NUMBER 4

risk management can benefit corporations in (1) the reduction of bankruptcy and distress costs, (2)the reduction of financing costs, and (3) the reduction of expected payments to stakeholders. However,some literature provides mixed evidence on whether risk management can increase firm value. MacKayand Moeller (2007) show that a concave cost function implicitly motivates oil companies to use morederivatives to hedge their financial and operational risks. With such a concave cost function, risk

management can increase firm value with simultaneous cost reduction and revenue enhancement.Following this line of study, we examine whether the cost function of insurance companies can bedepicted by concavity or convexity. In addition, we further explore whether a concave/convex costfunction of the insurance industry can increase/decrease insurers firm value while using the risk man-agement tools of financial derivatives and reinsurance. We use the degree of efficiency to measure aninsurers performance. The efficiency analysis for the insurance industry has focused on the impact ofmarket structures (Rai 1996; Choi and Weiss 2005), organizational forms, corporate governance andownership issues (Cummins et al. 2004), regulatory bodies and rules in different nations (Cumminsand Rubio-Misas 2006), and mergers and acquisitions (Cummins et al. 1999). In addition, Cumminset al. (2006) investigate the effect of risk management on insurance efficiency based on stochasticfrontier analysis. They assume that the risk management decision is the outcome of the macroeconomicand environmental factors, given that a firms objective function is to minimize the total cost. Fenn et

al. (2008) consider the volatility of cost efficiency in the European insurance industry, but the impactof risk management variables is not investigated.

This current paper differentiates itself from previous research by considering both reinsurance andfinancial derivatives in the analysis of cost efficiency. In addition, this study simultaneously investigatesthe effect on the mean and variance of cost efficiency from the managerial decisions on the utilizationof risk management tools. Furthermore, we make the first attempt to identify the concavity or convexityof insurers cost functions in order to create a linkage of insurers risk management incentives to firm

value creation. We intend to relate cost functions of different organizational forms of insurers (mutualvs. stock insurers) with different risk management strategies. The results of this study provide impor-tant insights for the practice of risk management in the P/L insurance industry.

The next section of this paper describes the one-step stochastic frontier model and the hypotheses.Section 3 describes the data and variables used in our analysis. Section 4 presents empirical results,

and the final section concludes.

2. MODEL SPECIFICATIONS AND HYPOTHESESThis paper adopts the stochastic frontier approach developed in Wang and Schmidt (2002) to considerhow risk management decisions and firm characteristics affect efficiency and cost function within onestep. The application of the stochastic frontier approach can further allow us to depict whether insurerscost functions carry the feature of concavity or convexity. McKay and Moeller (2007) conclude that aconcave cost function can enhance firm value from hedging, while other literature provides mixedevidence. We make the first attempt to examine the functional form of P/L insurers cost functionsand to investigate whether the conclusions provided in McKay and Moeller (2007) is supported in theP/L insurance industry. Compared to the one-step stochastic frontier approach, the most commonly

applied Data Envelopment Analysis (DEA) model (nonparametric) in the existing efficiency literature(e.g., Noulas et al. 2001; Jeng and Lai 2005) cannot depict the shape of cost functions and thereby isunable to directly link cost function to risk management strategies. In addition, it has also been shownthat the estimates of parameters in the one-step procedure are statistically more efficient than thetraditional two-step approach (i.e., the simple stochastic frontier approach combined with Tobitregression).

The general form of the one-step stochastic frontier model is illustrated by the following equations:

ln CR CR(Q, P, Z, T) u , it it it

R Fu R F , it it it it 2 exp(R F ), u it itit

(1)(2)(3)


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EFFECTS OF RISK MANAGEMENT ON COST EFFICIENCY AND COST FUNCTION OF THE U.S. PROPERTY AND LIABILITY INSURERS 489

where ln(CRi,t) is defined as the natural log of total costs from underwriting and investment on theincurred basis for firm i at time t, and CR(Q, P, Z, T) represents the cost frontier (function) of aninsurer and renders the least cost given the level of insurance services Q, the input price P, control

variables Z, and time T. Specifically, Q and P represent respective output quantity and input pricesthat are related to underwriting and investing activities, Z depicts firm characteristics variables and

represents the organization form (mutual vs. stock insurers), and T controls for time effects withinthe sample period. In equation (1), is the normal residual term of the cost function and uit is definedas the one-sided cost inefficiency for firm i at time t.1 Equation (2) enables us to explicitly examinethe factors contributing to the degree of cost inefficiency uit. The factor Rit is defined as a vector of

variables that are related to risk management. F is defined as financial intermediate outputs and isspecified by the ratio of surplus to risk-based capital and the growth of net premium written. The riskmanagement variables R and financial intermediate outputs F are both determined by the managementof the insurance company, and their effects are depicted by the coefficients ofR and F. Here is theresidual term of cost inefficiency function. Equation (3) indicates how the variance of cost inefficiency(2) is affected by financial intermediate outputs variable F and risk management variable R, whichconsists of underwriting risks and investment risks.

The one-step stochastic frontier model considers not only the cost efficiency level but also its vari-

ance. In the model equations (2) and (3) capture how the utilization of risk management mechanismsimproves the mean and variance of cost inefficiency, respectively. Equations (1) and (2) indicate thedegree of the mean of cost efficiency enhanced through the implementation of risk management, andequations (1) and (3) identify the degree of the variance of cost efficiency that is improved via riskmanagement. The inclusion of a vector of variables that are related to risk management in equations(2) and (3) enables us to consider the separate effect from managing underwriting risks or investmentrisks. The inclusion of equations (1), (2), and (3) in the model takes the mean and variance of costefficiency into account simultaneously, which contributes to the literature by expanding the insuranceefficiency study to consider different perspectives from both shareholders (emphasizing efficiency en-hancement) and policyholders (averse to variance).

As indicated in the insurance efficiency literature (e.g., Cummins and Weiss 1998; Cummins et al.2006), the exact form of the cost function described in equation (1) is unknown, and without loss of

generality, a natural log functional form of cost function is applied for empirical applications. Equations(4), (5), and (6) below carry out the empirical applications based on the one-step stochastic frontiermodel described in equations (1), (2), and (3).2 The variables in equations (4), (5), and (6) are thesame as those defined in equations (1), (2), and (3). In addition, to examine whether the cost functionof the P/L insurance industry carries the concavity/convexity feature, in equation (4) we include quad-ratic terms of input prices and its interacted terms with output quantity. McKay and Moeller (2007)discuss the case of financial derivatives use. This current study examines whether the link betweenconcavity/convexity and value creation can be observed in both reinsurance and financial derivativesrisk management mechanisms. The equations are as follows:

Q Pln CR ln Q P it i v vit s sitv s

Q,P2 2

{(ln Q ) (ln Q ln Q ) (ln Q P ) (P ) (P P )} v vit vit vjt vit sit vit sit sjtvt s

D D u , t s it itt s

R Fu R F , it it it it

2 exp(R F ). u it itit

(4)

(5)

(6)

1 The cost inefficiency is represented by eu. Its numerical interpretation is as follows: for example, with a value of 1.02, the total cost is 2%

above the optimal level of the cost function. Thus, it is recognized as a measure for inefficiency level.2 For more details regarding total cost and cost efficiency, see Fu and Heffernan (2008) and Baumol et al. (1980).


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In addition, the variables T and Z in equation (1) are realized by the dummy variables Dt

and Ds

inequation (4), which are designed to control the year effect and the ownership effect, respectively. D

s

equals 1 for stock insurers. The specifications of the models imply that the outputs of property/liabilityinsurers are determined by the overall market or macroeconomic factors. Therefore, insurers shouldimprove cost efficiency to reduce total costs and hence increase profit. In other words, given the output

level of an insurer, the best strategy of the insurer is to improve cost efficiency or reduce total costs.Using reinsurance is to directly manage underwriting risks, and reinsurance premiums have to be

paid up front. It is likely that insurers can adopt alternative methods to manage underwriting risksthrough underwriting insurance in different lines or in different geographic areas. We argue that inthe long run, reinsurance is likely to reduce total costs, but, on the other hand, reinsurance costs canbe higher in the short run. Consequently the net effect of reinsurance on total costs can be positiveor negative. Nevertheless, insurers utilize reinsurance and expect to increase the mean of cost efficiencyfrom its net effect. As a result, we develop the following hypothesis:

H1:

The amount of reinsurance (R1) is positively associated with the mean of cost efficiency.

On the other hand, using financial derivatives is to directly manage investment risks to reduce fi-nancing costs as well as bankruptcy costs, as indicated in Smith and Stulz (1985). The evidence fromthe banking literature suggests that the use of derivatives is often referred to as off-balance sheetactivities, and Clarks and Siems (2002) show that the cost and profit efficiency can be improved viaoff-balance sheet activities. In addition, using sample banks from Taiwan and Latin America, Lieu etal. (2005) and Rivas et al. (2006) conclude that banks can enhance cost efficiency if they apply morederivatives. With similar characteristics between the banking and insurance industry, we hypothesizethat the use of derivatives can increase the mean of insurers cost efficiency:

H2:

The notional amount of derivatives (R2) contributes positively to the cost efficiency of insurers.

The variance of cost efficiency can be regarded as a risk factor. In other words, when the variance

of cost efficiency is large, the total cost of an insurer is more volatile. Intuitively, insurers tend to applyboth R

1and R

2to reduce uncertainty, namely, the variance of cost efficiency. Therefore, Hypothesis 3

suggests a negative relationship between risk management variables and the variance of cost efficiency:

H3:

The risk management variables (R1

or R2) are negatively related to the variance of cost efficiency.

3. DATA AND VARIABLESTo empirically examine how risk management affects the mean and variance of cost efficiency basedon the one-step stochastic frontier model, we collected insurance data from the National Association

of Insurance Commissioners (NAIC) database. In addition, the data of derivatives use by insurers areseparately retrieved from Schedule DB. Insurers are required to file regulatory annual statements andany derivatives transactions with the NAIC. The three-year sample period covers years 2002, 2003, and2004,3 each of which includes 1,654, 1,662, and 1,654 sample insurers, respectively. The total numberof observations over this three-year sample period is 4,970, of which 3,865 firms are stock insurers and1,105 firms are mutual insurers.

3 Due to limited funding support for the data purchase of the NAIC annual statements and the NAIC Schedule DB data, this study is limited

to include only a three-year sample period. This study will serve as a basis for future research that will expand to a longer sample period.


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This paper adopts the intermediary approach (Brockett et al. 2005) to specify the empirical costfunction.4 In particular, the dependent variable of the cost frontier in equation (4) is the incurred-basis total cost that includes investment expenses and underwriting expenses derived from underwritingnew policies, maintaining existing policies, and reserving.

To empirically implement the methodology of the one-step stochastic frontier approach, we collect

the variables that are related to output quantities (Q), input prices (P), risk management (R), andfirm characteristics (Z). Since investment and underwriting risks are the two major categories of risksthat insurance companies intend to manage, we collect the output variables and risk management

variables that are specifically related to investment and underwriting activities.In terms of output variables, Q consists ofQ

1and Q

2. Q

1is defined as the investment-related output

quantity measured by the ratio of net investment income to total assets. Q2

represents theunderwriting-related output quantity defined by the ratio of total loss incurred to net premium written.

With respect to the variables related to risk management R, it includes R1

and R2, which represents

reinsurance premium written (R1

) and the notional transactions amount of derivatives (R2),

respectively.The input price P is categorized based on the insurers ownership that can be classified by policy-

holders, stockholders, and debt holders. For the vector of input prices, P includes P1, P

2, and P

3. P

1

stands for the ratio of total dividend specifically paid to the stockholders and change in treasury stocksto the surplus; P

2is the surplus growth rate, defined as the ratio of current-year surplus to previous

surplus; and P3

denotes the ratio of interest expenses to surplus. For stock insurers, there exist P1, P

2,

and P3, while for mutual insurers only P

2and P

3are used. P

2captures the proxy of an input price that

stands for the contribution of surplus. Last, F factors, representing the exogenous firm characteristic variables, include the ratio of surplus to regulatory required risk-based capital ( Sup RBC) and thegrowth rate of net premium written ( NPW GW). The choice of Sup RBC and NPW GW is based onthe empirical work by Born et al. (2009).

Table 1 presents the summary statistics of the variables for the entire sample, stock insurers, andmutual insurers, which include 4,970, 3,865, and 1,105 observations, respectively. In addition, thecomparison between stock and mutual insurers is also provided in Table 1. It is shown that, on average,the total cost CR for the entire P/L insurance industry in the period from 2002 to 2004 is about

$189,421 million, and $220,225 million and $151,641 million for stock and mutual insurers, respec-tively. The difference in total costs between stock and mutual insurers is not significant. As shown,each variable shows a greater degree of variability, evidenced by a larger standard deviation and largercoefficient of variation (CV). For example, the CV of the total cost for the entire sample is about 5.32.Mutual insurers tend to show larger variability of total costs than stock insurers. The CV is 4.92 forstock insurers, while it is 7.12 for mutual insurers. Results based on the univariate comparison suggestthat the total cost of mutual insurers is more volatile than that of stock insurers in the sample despiteinsignificant mean difference.

In terms of the output variables related to investing (Q1) and underwriting activities (Q

2), stock

insurers, on average, show a significantly larger net investment income than mutual insurers, withmean values of $33,092 million and $3,397 million, respectively. However, no significant difference is

found in the underwriting outputs between stock and mutual insurers. The coefficients of variation ofQ1

and Q2

for stock and mutual insurers suggest a different degree of variability in investing andunderwriting activities. For stock insurers, the variability of underwriting outputs (with CV 42.28)is greater than that of investing outputs (with CV 29.71). On the other hand, for mutual insurers,the variability of investing outputs is greater with the value of CV at 9.92 compared to 2.67 for un-derwriting outputs.

4 Brockett et al. (2005) indicate that the intermediary approach is more relevant than the value-added (production) approach in the insurance

industry.


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Table 1

Summary Statistics and Comparisons

Panel I Summary Statistics and Comparisons

All Insurers Stock Insurers Mutual Insurers

Two-Sample ZTest between

Stock andMutual Insurers

Variable Mean S.D. C.V. Mean S.D. C.V. Mean S.D. C.V. H0:

s

m 0

CR 189,421 1,007,500 5.32 200,225 985,870 4.92 151,641 1,079,340 7.12 0.84Q

126,488 866,824 32.73 33,092 982,748 29.70 3,397 33,713 9.92 1.87*

Q2 2,432 117,045 48.13 3,110 132,749 42.68 64 171 2.67 1.43

P1 (%) 3.87 25.70 6.64 P2

(%) 1.55 28.24 18.22 1.17 0.54 0.46 2.90 59.89 20.65 0.56P3 (%) 2.03 67.99 33.49 0.83 1.44 1.73 6.23 144.12 23.13 0.72

R1 163 800 4.91 186 860 4.62 84 534 6.36 3.30***

R2 1,353 95,194 70.36 1,739 107,951 62.08 3 63 21.00 1.00SUP RBC 0.96 0.23 0.24 0.97 0.19 0.20 0.90 0.31 0.34 4.15***NPW GW 5.77 183.05 31.72 7.07 207.57 29.36 1.22 1.39 1.14 1.75*TA 649,512 3,010,270 4.63 667,853 2,566,674 3.84 585,363 4,209,905 7.19 0.37

Panel II Number of Observations

Year Number of Firms Number of Firms Number of Firms

2002 1,654 1,281 3732003 1,662 1,296 3662004 1,654 1,288 366Total 4,970 3,865 1,105

Note: S.D. is the sample standard deviation. The two-sample Z test is to test the null hypothesis H0: mutual. *, **, and *** indicate thestocksignificance at the 10%, 5%, and 1% levels, respectively.R1: premium written of reinsurance in million dollars

R2: notional amount of derivatives in million dollars

CR: total costs in millions of dollarsQ

1: investment related output quantity, i.e., net investment income in millions of dollars

Q2: underwriting related output quantity, i.e., total loss incurred in millions of dollars

P1: the ratio of the sum of dividend paid to the stockholders and change in treasury stocks to the surplus, i.e., (dividend paid change intreasury stocks)/surplusP2: the surplus growth rate, i.e., surplus in the current year/surplus in the previous year

P3: the ratio of interest expenses to surplusSUP RBC: the ratio of surplus to RBCNPW GW(%): the growth rate of net premium writtenTA: total assets in million dollars

Regarding the input price variables, we observe that on average P2

and P3

for stock insurers aresmaller than those for mutual insurers, but the difference is not significant. For example, the valuesofP

2and P

3for stock insurers are 1.17% and 0.83%, whereas they are 2.90% and 6.23% for the mutuals.

In terms of the utilization of risk management, as shown in Table 1, compared to mutual insurers,stock insurers tend to apply a larger degree of reinsurance and financial derivatives for managingunderwriting and investment risks. The difference in reinsurance is significant between stock and mu-tual insurers, whereas no significant difference is seen in the use of derivatives. Specifically, the average

values ofR1

and R2

for stock insurers are $186 million and $1,739 million, respectively, and they are$84 million and $3 million for mutual insurers. This result suggests that stock insurers implement riskmanagement tools for both underwriting and investment risks more intensively than mutual insurers.In addition, stock insurers show a smaller variability in reinsurance use than mutual insurers with therespective values of CV of 4.62 and 6.36. On the other hand, stock insurers use of financial derivativesshow a greater degree of variability, with a CV value of 62.28 compared to that of mutual insurers witha value of CV at 21. The comparison of firm characteristics shows that the ratio of surplus to RBC andthe premium growth rate of stock insurers are significantly greater than those of mutual insurers. Onthe other hand, no significant difference in firm size is observed between stock and mutual insurers.

Generally speaking, each input, output, or firm characteristic variable of stock insurers does notconsistently show greater variability than that of mutual insurers. With a different degree of variability


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in the variables, it is likely that the cost functions of stock and mutual insurers can be different,suggesting their different applications of risk management mechanisms for underwriting and investingactivities.

4. EMPIRICAL RESULTS AND ANALYSESThe empirical results of the applications of the one-step stochastic frontier model are presented inTables 2, 3, and 4. In particular, the results of cost function analyses as indicated in equation (4) aresummarized in Table 2. These analyses further associate the concavity or convexity of the cost frontier

with the hedging strategy of an insurer. Tables 3 and 4 report the results of cost efficiency analysesfrom the utilization of different risk management mechanisms for underwriting and investing risks asindicated in equations (5) and (6). As suggested in Wang and Schmidt (2002), the simultaneous con-sideration of the mean and variance of cost efficiency enables us to detect how risk managementutilization affects the mean and variance of cost efficiency.

4.1 The Analysis of Cost Function

Table 2 presents the estimation of the cost functions for the entire sample and the subsamples of stock

and mutual insurers. In order to identify the effect of firm size (measured by total assets), for eachsubsample, we further divide it into subgroups of big and small insurers. For the entire sample, its costfunction incorporates all the observations of stock and mutual insurers. The results indicate that amajority of coefficients are statistically significant (19 out of 24). For example, ln Q

1is negatively

associated with CR, while ln Q2, P

1, P

2, and P

3show positive effects. In addition, the model parameters

and are significant at the 1% level. Lambda is defined as the ratio of the standard deviation of costinefficiency to that of the random error term, that is,

u/

. A significant suggests a significant

difference of cost inefficiency from the random error. Sigma is defined as the composite standard

deviation of both cost inefficiency and random error: A significant implies that the total2 2 .u

composite error term significantly departs from the cost function. Consequently the results of signifi-cant of and indicate the existence of cost inefficiency.

The analysis of the cost function can be associated with the relation between the concavity of thecost frontier and the hedging strategy of a firm (McKay and Moeller 2007). For the entire sample (P

1)2,

(P2)2, and (P

3)2 are all negatively significant at the 1% level, which suggests that the cost function for

the P/L industry is concave along the dimensions of P1, P

2, and P

3, respectively. The year dummies

(D2003, D2004) are included to control the year effect. The insignificant coefficients of year dummiessuggest no difference over the sample period. The stock insurer dummy variable shows a positive andsignificant effect on total costs at the 1% level, which suggests that stock insurers tend to have highertotal costs compared to mutual insurers, which is consistent with the univariate analysis shown inTable 1.

As for the subsamples of stock and mutual insurers, the results document that the estimates of stockinsurers are principally consistent with those observed in the entire sample. In particular, the coeffi-cients of the squared term of the surplus growth rate (P

2)2 of both stock and mutual insurers are

negatively significant at the 10% and 5% levels, respectively. In addition, the coefficient of the interestexpense ratio (P3)2 of mutual insurers is negatively significant at the 10% level. The results suggest

that the concavity relative to the surplus growth rate remains in both organizational forms, while theconcavity relative to the interest expense ratio holds only in the mutual insurers group.

On the other hand, compared to the results of stock insurers and the entire sample, the results ofthe mutual insurers group present different effects on their cost functions. For instance, the coefficientof ln Q

1in the mutual subsample is positive, whereas it has negative effects in the entire sample and

stock insurers subsample. This suggests that while investment income of the entire P/L industry andthe stock insurers can significantly reduce total costs, the investment income of the mutual groupincreases its total costs. In addition, ln Q

2has positive effects on the cost function for the entire

sample and stock insurers subsample, whereas we observe insignificant effects in the mutual subsample.


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Table 3

Effects of Risk Management Variables on the Mean of Cost Inefficiency: Equation (5)

All Observations Stock Small Stock Big Stock

Variable Coefficient t value Coefficient t value Coefficient t value Coefficient t value

Constant

5.78439

1.42 0.11509 0.07

0.75740

0.02 0.00001 0.00SUP RBC 0.02548 0.03 0.10153 0.08 0.25791 0.01 0.00001 0.00NPW GW 0.00381 0.64 0.00121 0.89 0.01880 1.26 0.00001 0.00R1 0.00126 5.84*** 0.00133 17.31*** 2.06342 8.54*** 0.00076 6.32***

R2 0.00003 4.07*** 0.00000 0.08 0.00011 0.08 0.00000 0.07

Mutual Small Mutual Big Mutual

Variable Coefficient t value Coefficient t value Coefficient t value

Constant 6.24172 0.60 3.61994 9.07*** 1.36182 12.85***SUP RBC 0.82205 0.31 4.51933 4.05*** 1.25728 5.64**NPW GW 0.03406 0.04 0.13716 0.19 1.59949 3.96**R1 0.00195 1.95* 0.00119 0.00 0.00050 0.00

R2 0.08105 0.96 0.00021 0.01 0.00334 0.00

Note: *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. Big (small) stock insurers are defined as those insurerswith firm size greater (less) than median size in the corresponding organizational group.R1: premium written of reinsurance in millions of dollars

R2: notional amount of derivatives in millions of dollarsSUP RBC: the ratio of surplus to RBCNPW GW(%): the growth rate of net premium written

Table 4

Effects of Risk Management Variables on the Variance of Cost Inefficiency: Equation (6)

All Observations Stock Mutual

Coefficient t value Coefficient t value Coefficient t value

Constant 7.96474 88.90*** 7.98192 72.11*** 7.49455 38.83***SUP RBC 0.00000 0.00 0.00000 0.00 0.05964 0.45NPW GW 0.00000 0.00 0.00000 0.00 0.00512 0.07R1 0.00019 7.20*** 0.00020 6.85*** 0.00111 7.13***

R2 0.00192 6,453.67*** 0.00219 7,351.91***

0.00185

2.51***

Note: *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively.R1: premium written of reinsurance in millions of dollarsR2: notional amount of derivatives in millions of dollarsSUP RBC: the ratio of surplus to RBCNPW GW(%): the growth rate of net premium written

Table 2 also presents a comparison based on firm size for each organizational form. No significanteffects of the squared terms of input prices, (P

1)2, (P

2)2, and (P

3)2, on cost function are observed for

different firm sizes.In summary, the direction of the impact of the output variables and input prices on cost functions

is generally the same as we expected for the entire P/L industry. In addition, the negative and signif-

icant coefficients of the squared input prices suggest concavity of the cost function, and thereby theP/L insurance industry tends to hedge risks for firm value creation purpose. The next step is to analyze

whether both risk management mechanisms can effectively increase firm value measured by efficiency.

4.2 The Analysis of Cost Efficiency

The analysis of cost efficiency here is to investigate the impact of risk management on the mean and variance of cost inefficiency. In particular, equations (5) and (6) test the significance of the contri-bution of the risk management utilization to the mean and variance of cost inefficiency. Negativeestimates of the coefficients of risk management variables in equation (5) suggest that the implemen-tation of risk management tools can decrease the mean of cost inefficiency, that is, increase the mean


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of cost efficiency. In addition, negative coefficients of risk management variables in equation (6) sug-gest a decrease in the variance of cost inefficiency.

Table 3 demonstrates the empirical results of equation (5) for the entire sample as well as for bothstock and mutual insurers. It displays the estimation of equation (5), indicating the impact of riskmanagement factors R on the mean of cost inefficiencyu.

As shown, for the entire sample, the coefficient of reinsurance use (R1) is positively associated withcost inefficiency (u), whereas the coefficient of derivatives use (R2) presents a negative relation withcost inefficiency. Both are significant at the 1% level. This indicates that for the whole P/L insuranceindustry, only the use of financial derivatives enhance cost efficiency, but not the use of reinsurance.The results are consistent with Hypothesis 2, but not with Hypothesis 1, in which we expect costefficiency enhancement through reinsurance. Neither stock nor mutual insurers show efficiency en-hancement through reinsurance.

Nevertheless, incorporating the concavity of the cost function into consideration of hedging imple-mentation, we observe that for the entire sample, the concavity of input prices (i.e., cost) does con-tribute to the use of financial derivatives, thereby leading to enhancement of cost efficiency. The resultson the use of financial derivatives are consistent with the conclusions in McKay and Moeller (2007).Differently from McKay and Moeller, our research also examines the relationship between reinsurance

and cost efficiency and documents a negative effect of reinsurance on cost efficiency. No significantdifference is observed between small and big insurers.

Table 4 depicts how risk management variables are associated with the variance of cost (in)efficiencyas indicated in equation (6).5 For the entire sample, both R1 and R2 are positively and significantlyrelated to the variance of cost inefficiency, which suggests that either the use of reinsurance or theuse of derivatives increases the volatility of cost efficiency. As for insurers different organizationalforms, the sample of stock insurers group shows consistent results with those of the entire sample.Nevertheless, for mutual insurers, the implementation of financial derivatives decreases the volatilityof cost efficiency as expected. For stock insurers, ownership is dominated by public shareholders, whomay desire a larger degree of volatility to increase value. As a result, a positive effect of financialderivatives implementation on volatility is observed.

5. CONCLUSIONSMore and more enterprises sense the importance of risk management. As for the insurance industry,there are two major types of risks: underwriting risks and investment risks. An insurance company canreduce its risks either by purchasing reinsurance policies or by using financial derivatives. This paperadopts the one-step stochastic frontier approach to explore the impact of risk management tools suchas derivatives and reinsurance on both cost function and cost efficiency in the U.S. property-liabilityinsurance industry. This study takes the first attempt to examine the cost function of the insuranceindustry in order to link the cost function concavity (or convexity) to risk management and then tofirm value enhancement

According to the empirical results, the cost function for the entire P/L insurance industry carriesthe concavity feature not only toward the equity related input price, but also toward the debt-related

input prices; and the P/L insurers tend to hedge by using equity-related derivatives, such as financialderivatives. As to the effect of derivatives on cost efficiency, the empirical result documents that theuse of derivatives significantly enhances cost efficiency for the entire P/L insurance industry. On theother hand, the use of reinsurance does not enhance cost efficiency for the P/L insurance industry. As

we take one step further to examine the effects of risk management on the volatility of cost efficiency,

5 There is no significant difference observed between small and big insurers. Therefore, the comparison results between small and big insurers

are not presented in Table 4.


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we observe that for the entire P/L industry and for the sample of stock insurers, the use of derivativesincreases the volatility of cost efficiency, resulting in a more volatile cost efficiency. Theoreticallyinsurers can adopt derivatives to hedge the investment risk. However, when insurers aggressively takeon risks to seek higher investment reward, the volatility of cost efficiency will increase to closely align

with stockholders interests: That is, the use of derivatives is likely to increase the risk of the insurer

and defeat the purpose of risk management. The implication of this finding is that insurers shouldkeep a balance between the efficiency gain and efficiency volatility with the use of derivatives.

6. ACKNOWLEDGMENTS

The authors thank Professor Mary Hardy and the anonymous referee for their valuable and insightfulcomments. Hong-Jen Lin is grateful for the research award offered by the Research Foundation of theCity University of New York (PSC-CUNY Award no. 63415-00 41).

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Discussions on this paper can be submitted until April 1, 2012. The authors reserve the right to reply to anydiscussion. Please see the Submission Guidelines for Authors on the inside back cover for instructions on thesubmission of discussions.

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Naaj 2011 Vol15 No4 Lin Wen Yang