Determinants of the Demand for Reinsurance for the … Review Vol. 34 (Oct 2015), 125-138 Vincent Y....

Management Review

Vol. 34 (Oct 2015), 125-138

Vincent Y. Chang

Determinants of the Demand for Reinsurance

for the U.S. Property-Liability Insurance Industry:

Quantile Regression Analysis

(Received Feb 19, 2013; First Revision Dec 11, 2013; Second Revision Apr 9, 2014;

Third Revision Jun 6, 2014; Accepted Aug 7, 2014)

1. Introduction

Reinsurance is an important risk management device for primary insurers, especially when that risk leads to catastrophies such as the 921 earthquake in Taiwan, Hurricane Katrina in the U.S., and the earthquake in Sichuan province, in China. Insurers always treat reinsurance as a hedge mechanism to overcome non- diversifiable risks. In addition, reinsurance can be used to limit the insurer’s underwriting risks and reduce the probability and expected costs of potential bankruptcy. Thus, reinsurance is often regarded as an in- dispensable and effective risk management mechanism for primary insurers.

It is critical for property-liability insurers to purchase sufficient reinsurance contracts. The short- term contract characteristic forces managers to pay more attention to a firm’s liquidity. Because reinsure- ance can be treated as a substitute for firm liquidity, as well as for firm capital, property-liability insurers actually rely on reinsurance to reduce short-term financial pressures and liquidity risks. Furthermore, short-term insurance policies may generate insufficient cash reserves, encouraging managers to demand further reinsurance to mitigate the impact of catastrophic risk on the near-event horizon.

The current literature provides insightful evidence of factors for determining insurers’ demand for reinsurance (Carneiro and Sherris 2005; Chang and Jeng 2015; Cole and McCullough 2006; Garven and Lamm-Tennant 2003; Kader and Adams 2007; Mayers and Smith 1990; Shiu 2011; Wang et al. 2008; Yanase 2015). However, most of analyses adopt the ordinary least squares regression (OLS), or the two-stage least squares regression (2SLS) methods, which model the relationship between the covariates of firm-specific characteristics and the conditional mean of demand for reinsurance, by examining issues related to demand for reinsurance. To the best of my knowledge, rela- tively few papers have studied the determinants of an insurer’s demand for reinsurance using a quantile regression (QR) approach or a two-stage quantile regression approach (2SQR), even though QR could pro-

Vincent Y. Chang is an Assistant Professor of Depart-

ment of Insurance & Department of Finance (joint ap-

pointment), Chaoyang University of Technology. Email:

[email protected]. The author gratefully acknowl-

edges the two anonymous reviewers for their construc-

tive comments and the financial support of the Ministry

of Science and Technology (National Science Council)

(MOST (NSC) 100-2410-H-324-010). The author would

like to thank Enago (www.enago.tw) for the English lan-

guage review.

vide an alternative dimension to the study of demand analysis and contribute to the analysis of heavy-tailed data

1. Thus, this study aims to fill an important void

in the literature.

It is predicted that insurers with higher levels of reinsurance tend to be safer and experience less financial pressure and liquidity risk than insurers with lower reinsurance levels. Thus, one can predict that the insurer’s behavior, for instance, including risk behavior and financial and operating strategies, should present a significant difference in the lower and higher reinsurance quantiles. Thus, the QR approach natu- rally provides useful functions for examining the determinants of insurer’s demand for reinsurance across various quantiles, especially for insurers within the lower quantiles.

Prior studies suggest that insurer leverage and

liquidity maintenance may represent endogenous in-

fluences on demand for reinsurance (Chang and Jeng

2015; Shiu 2011)2. Therefore, in this study, the 2SQR

paradigm is implemented to correct for endogeneity

bias (Amemiya 1982; Chen and Portnoy 1996; Kim

and Muller 2004; Powell 1983)3. Instead of focusing

on the average effects of covariates on the demand

for reinsurance, this study applies the 2SQR approach

to explore the potentially differential effects across

the demand distribution. Using pooled, time-series, and

1Koenker and Bassett (1978) introduce the QR approach.

Koenker and Hallock (2001) further propose that there

are several advantages to this approach. For example,

the QR approach seizes the critical features of the data-

base used and offers a complete picture of the covariate

effect when a set of percentiles is modeled (Chang and

Tsai 2014). 2I sincerely thank the anonymous referee who proposed

that this study should provide the endogeneity test a-

mong demand for reinsurance, leverage, and liquidity. In

this study, the Hausman’s specification test indicates that

the endogeneity influences of insurer’s leverage and li-

quidity on demand for reinsurance do exist. This study de-

tails the endogeneity test in the methodology section. 3The QR estimation bias due to endogenous variables is

as a result of ignoring the dependence between the ex-

planatory variable and the unobserved error term. Since

the error term of reduced-form of the endogenous vari-

able is correlated with the structural error term of the de-

pendent variable, it can be regarded as an omitted con-

trol variable in the traditional QR model. The QR esti-

mation has large MSE (mean square errors) as the de-

gree of endogeneity is higher. Thus, this study uses 2SQR

to correct for the endogeneity bias from endogenous var-

iables.

http://www.enago.tw/

Management Review, October 2015 126

cross-sectional data for U.S. property-liability insurers from 2006 to 2010, the 2SQR results show that the influence of some firm-specific characteristics, for instance liquidity and loss development, are not constant across quantiles, which contradicts the findings of prior studies (Chang and Jeng 2015; Cole and Mc- Cullough 2006; Hau 2006; Wang et al. 2008). The evidence presented here indicates that 2SQR analysis could provide a richer and more insightful explanation for the lower and higher quantiles of the rein- sureance distribution than by just using the 2SLS approach. In addition, the evidence indicates that other firm-specific characteristics (e.g., leverage, profitability, organizational form, and concentrations) exert an impact on the consistence of results with the 2SLS approach. However, the magnitude of these influences on demand for reinsurance differs significantly across various reinsurance quantiles. These findings again enhance the main argument proposed in this study.

The contributions of this study are as follows. First, this study is the first article to analyze the determinants of the insurer’s demand for reinsurance within a particular quantile of the overall demand distribution, and it provides more meaningful information than the traditional 2SLS approach, while also com- pensating for the inherently biased estimations of the latter model. Thus, the present study fills an important analytic gap in the literature by applying the 2SQR analysis. Second, the 2SQR approach can analyze the influence of covariates at a specific point in the reinsurance distribution, which provides more insightful evidence for the lower and higher quantiles of the demand distribution than the 2SLS approach. Specif- ically, this study finds that liquidity and the loss of development are not constant across various quantiles, which contradicts the findings reported in previous studies (Chang and Jeng 2015; Cole and McCullough 2006; Hau 2006; Wang et al. 2008). In addition, the magnitude of the impacts for other firm-specific characteristics on demand for reinsurance varies signify- cantly across the quantile range, even if the signs of empirical results are consistent with the previous literature. The evidence further concurs with the main hypotheses, which are developed and presented below. Third, the insightful information obtained using the 2SQR approach provides an incentive for policyholders, policy makers, and/or regulators to assess the insurer’s demand for reinsurance through these firm- specific characteristics, especially in the case of insurers in the lower reinsurance quantiles.

The rest of this paper is structured as follows. Section II describes the primary hypotheses and variables used in this study. Section III briefly discusses the 2SQR approach. Section IV interprets the data, presents model fitness checks, and reports the empirical results. Section V concludes the article.

2. Hypotheses Development and Variables Description

2.1 Hypotheses Development

Note that the expectations of all exogenous variables for the traditional 2SLS approach may be distinct from the 2SQR approach. For instance, the hy-

potheses of bankruptcy cost, agency cost, risk bearing, and renting capital all propose that a higher- leverage insurer tends to increase their demand for reinsurance (Chang and Jeng 2015; Cole and McCul- lough 2006; Garven and Lamm-Tennant 2003; Shiu 2011; Wang et al. 2008). However, the impact of leverage on demand for reinsurance may differ between insurers in lower and higher quantiles. Since the negative marginal effect of reinsurance on underwriting risk is likely to diminish in the higher reinsurance quantiles, the effect of leverage on reinsurance is likely to increase as the level of reinsurance increases. For each additional unit of leverage, an insurer with a high level of reinsurance will need “more” reinsure- ance to maintain its underwriting risk at a target level. In contrast, the inverse argument will be presented regarding the lower reinsurance quantiles. For each additional unit of leverage, an insurer with a lower level of reinsurance will need “less” reinsurance to satisfy its target level of underwriting risk, because the negative marginal effect of reinsurance on underwriting risk is likely to increase for the lower reinsurance quantiles.

Furthermore, the substitute hypothesis indicates that higher liquidity insurers tend to demand less reinsurance (Chang and Jeng 2015; Hau 2006). How- ever, it is predicted that the impact of liquidity maintenance on demand for reinsurance in the lower quantiles also differs from demand in the higher quantiles. Likewise, since the negative marginal effect of reinsurance on underwriting risk is likely to lessen in the higher reinsurance quantiles, the effect of liquidity maintenance on reinsurance is likely to decrease in the higher quantiles. Thus, for each increment of liquidity maintenance, an insurer with a high level of reinsurance will need “less” reinsurance to uphold its target level of underwriting risk. On the contrary, based on the argument similar to those above, to sus- tain its target level of underwriting risk, an insurer with a low level of reinsurance will need “more” reinsurance for each increment of liquidity maintenance in the lower reinsurance quantiles. The reason is that insurers in the lower reinsurance quantiles encounter more financial pressures or bankruptcy costs. This forces insurers to gradually increase their demand for reinsurance, even if they possess liquid assets. Accord- ingly, it is predicted that the determinants of demand for reinsurance across the reinsurance distribution are also different for other explanatory variables

4. Conse-

quently, this study sets out to test the following hypotheses:

Hypothesis 1: The impact of firm-specific charac-

teristics on demand for reinsurance

is different for firms at opposite ends

of the demand distribution.

Hypothesis 2: The predictions of determinants of re-

insurance differ between the 2SLS

approach and 2SQR approach.

4Overall, one can surmise that the influence of other ex-

planatory variables (e.g., organizational form, profitabi-

lity, tax issues, and loss of reserves) on demand for rein-

surance is also distinct at the lower and higher reinsure-

ance quantiles.

Reinsurance: Quantile Regression Analysis 127

The traditional analysis of demand determinants

employs the OLS/2SLS approach (Chang and Jeng

2015; Cole and McCullough 2006; Garven and Lamm-

Tennant 2003; Mayers and Smith 1990; Shiu 2011;

Wang et al. 2008; Yanase 2015). The 2SLS approach

relies on an a priori distributional assumption of the

dependent variable. In addition, the 2SLS approach

always accepts a homogeneous influence of the de-

pendent variable, which increases the estimation bias,

especially when the dependent variable is heterogene-

ous.

In contrast to the 2SLS approach, the QR/2SQR

approach, which presents a more complete picture of

the covariate effect, allows us to examine the differ-

ential effects across the demand for reinsurance dis-

tribution when a set of percentiles is modeled. Thus,

one could conduct an assessment at the tails of the de-

pendent variable by identifying the determinants sep-

arately. In addition, the 2SQR approach is robust and

less sensitive to the presence of outliers or skewed

tails. To sum up, the 2SQR approach is more effi-

ciently provide insightful information than the 2SLS

approach, especially at the tails of the reinsurance dis-

tribution. This study then hypothesizes that

Hypothesis 3: The 2SQR approach complements the

2SLS approach in terms of providing

efficient estimation and insightful in-

formation.

2.2 Description of Variables

Prior studies always adopted the reinsurance ratio as a proxy for the insurer’s demand for reinsure- ance (Chang and Jeng 2015; Cole and McCullough 2006; Garven and Lamm-Tennant 2003; Mayers and Smith 1990; Shiu 2011; Wang et al. 2008; Yanase 2015). In accordance with prior research, the insurer’s reinsurance ratio (Reins), which is defined as (affiliated reinsurance ceded + nonaffiliated reinsure- ance ceded)/(direct business written plus reinsurance assumed), is also used.

The literature suggests that several firm-specific characteristics, such as bankruptcy, tax issues, profitability, concentration level, organizational form, liquidity risk, solvency concerns, and business mix, may affect the insurer’s demand for reinsurance (Chang and Jeng 2015; Cole and McCullough 2006; Garven and Lamm-Tennant 2003; Mayers and Smith 1990; Shiu 2011; Wang et al. 2008; Yanase 2015). Accord- ingly, these firm-specific characteristics and hypotheses are then introduced as follows.

With regard to bankruptcy characteristics, three variables are adopted: firm size, leverage, and loss development. In accordance with the hypotheses of bankruptcy cost, agency cost, risk bearing, and renting capital, leverage is predicted to be positively related to the demand for reinsurance (Chang and Jeng 2015; Cole and McCullough 2006; Garven and Lamm- Tennant, 2003; Shiu, 2011; Wang et al. 2008). It is predicted that insurers writing more business relative to their surpluses should have a higher insolvency probability; therefore, greater leverage tends to produce a

higher demand for reinsurance. The measurement of the insurer’s leverage is defined as the direct business written to surplus (Leverage)

5.

As suggested in previous studies (Chang and Jeng 2015; Christensen, Hoyt, and Paterson 1999; Cole and McCullough 2006; Gaver and Paterson 1999; Grace 1990; Petroni 1992; Shiu 2011; Wang et al. 2008; Weiss 1985), potential financial constraints also play an important role in demand for reinsurance. Insurers with a positive loss development (i.e., under-loss reserving) will purchase more reinsurance contracts to surmount their potential financial constraints, whereas insurers will purchase less if they have a negative loss development (i.e., over loss reserving). In addition, Harring- ton and Danzon (1994) indicate that insurers may hide their underreported claim liability and capital inadequacy by using reinsurance. To sum up, the positive relationship between loss reserve and demand for reinsurance is expected. Following Cole and McCul- lough (2006), two-year loss development (2_years_ loss), which is defined as the development in estimated losses and loss expenses incurred 2 years before the current year and prior year (scaled by policyholders’ surplus), is chosen to proxy the insurer’s potential financial constraints.

Hau (2006) proposed that an insurance contract can provide additional liquidity to a nonfinancial firm, which implies that the relationship between corporate insurance demand and liquidity is substitutive. Based on his argument, this study extends his rationale to the insurance industry and argues that reinsurance contracts provide extra liquidity for primary insurers. In addition, insurers with more liquid assets have an increased ability to mitigate unexpected and/or emergent demand for cash, thus reducing their demand for reinsurance (Chang and Jeng 2015). Therefore, it is predicted that the insurer’s liquidity is negatively related to its demand for reinsurance. The liquidity (Liq) measurement is defined as the sum of cash and invested assets divided by total assets.

To reduce the probability of insolvency, small insurers tend to purchase more reinsurance contracts than large insurers (Chang and Jeng 2015; Cole and McCullough 2006; Garven and Lamm-Tennant 2003; Hoyt and Khang 2000; Mayers and Smith 1990; Shiu 2011; Wang et al. 2008; Weiss and Chung 2004; Yan- ase 2015). Thus, a negative relationship between firm size and demand for reinsurance is predicted. In this study, the natural logarithm of total assets is used to measure firm size (Size).

5As the anonymous referee points out, many prior stud-

ies also use liabilities to assets as a proxy for leverage.

This study also use liabilities to assets to reexamine the

2SQR for robustness check. Overall, the results are con-

sistent to the main findings of Table 3. This study does

not tabulate those specific results here. For a consistent

comparison, in this paper we follow previous studies

(Chang and Jeng 2015; Cole and McCullough 2006;

Garven and Lamm-Tennant 2003; Shiu 2011; Wang et al.

2008) and use the ratio direct business written to surplus

to measure the insurer’s leverage. I sincerely thank the

anonymous referee for pointing out this important issue.


This study also uses tax-exempt investment in-

come relative to total investment income (Tax_ex) as

a proxy to capture the influence of expected tax liabil-

ity and/or tax-favored assets. Smith and Stulz (1985)

and Mayers and Smith (1990) indicate that insurers

can reduce their earnings volatility through reinsure-

ance purchasing, thereby reducing their expected tax

liability. Garven and Lamm-Tennant (2003) propose

that insurers can benefit from reinsurance purchasing

in terms of mitigating the costs of huge unexpected

losses and enhancing investments in tax-favored as-

sets. Thus, a positive relationship between the tax-

exempt factor and the demand for reinsurance is pre-

dicted. However, the empirical findings of Adams, Hard-

wick, and Zou (2008), Graham and Rogers (2002),

and Shiu (2011) do not support the positive influence

of tax-exempt factors on demand for reinsurance. In

addition, Adams, Hardwick, and Zou (2008) and Shiu

(2011) further propose that insurers with a higher mar-

ginal tax rates will purchase less reinsurance to de-

crease expected tax liabilities. To sum up, the ex-

pected sign between the tax-exempt factor and the de-

mand for reinsurance is mixed.

Insurers with higher profits tend to demand less

reinsurance because they have higher payment capac-

ity and cash flows to confront loss claims and finan-

cial pressures (Chang and Jeng 2015; Cole and Mc-

Cullough 2006; Mayers and Smith 1990; 2004; Pow-

ell and Sommer 2007; Shiu 2011; Wang et al. 2008).

To control for profitability impacts on demand for re-

insurance, the insurer’s profitability, that is, the in-

surer’s return on assets (ROA), is adopted, which is

defined as net income plus tax and interest expenses,

divided by total assets.

The degrees of business concentration (Bus_H)

and geographic concentration (Geo_H) also influence

the insurer’s reinsurance decision. The business Her-

findahl Index is defined as the sum of the squares of

the ratio of the dollar amount of direct business writ-

ten in a particular line of insurance to the dollar a-

mount of direct business across all 26 lines of in-

surance; in addition, the geographic Herfindahl Index

is defined as the sum of the squares of the ratio of the

dollar amount of direct business in state j to the total

amount of direct business across all states (Chang and

Jeng 2015; Cole and McCullough 2006; Kim, Mayers,

and Smith 1996; Mayers and Smith 1990; Shiu 2011;

Wang et al. 2008). The argument for risk diversify-

cation indicates that insurers with a higher concen-

tration in a given line of business, or in a given geo-

graphic area, may have higher incentives to purchase

more reinsurance. In contrast, the real services hy-

pothesis indicates that reinsurers not only provide

protection against unexpected losses but also supply

real services via specialized knowledge and econom-

ies of scale. Thus, to obtain more technical support

from reinsurers, less business- or geography-concen-

trated insurers may demand more reinsurance. In

addition, Shiu (2011) indicates that more concentrated

insurers could specialize in, and may underwrite less

volatile lines of business, or less-risky contracts and,

therefore, demand less reinsurance. In sum, the pre-

dictions between business and geographic concentra-

tions and the demand for reinsurance are undetermined.

Since the organizational form hypothesis indi-

cates that stock insurers can benefit from diversify-

cation (outside shareholders) and/or from a lower cost

of raising external capital, this suggests that stock

insurers tend to purchase less reinsurance than mutual

insurers (Mayers and Smith 1990). On the contrary,

the agency cost hypothesis proposes that insurers’ pur-

chases of more reinsurance could mitigate the under-

investment problem among residual claimholders (Ad-

ams 1996; Shiu 2011). Consistent to the agency cost

hypothesis, Adams (1996) and Shiu (2011) find that

stock insurers purchase more reinsurance than mutual

insurers. Summing up, this study proposes that the pre-

diction effect of the organizational form is mixed. The

stock dummy variable (Stock) is used to control for

the organizational form effect, which is defined to be

1 if the insurer is the stock insurer, and 0 otherwise.

To control for the systematic difference between

a single firm (nonaffiliated insurer) and a firm be-

longing to a group (affiliated insurer), the single dum-

my variable (Single) is used. It equals 1 if the insurer

is nonaffiliated, and 0 if it is affiliated. A negative

relationship between the single dummy variable and

the demand for reinsurance is expected because insur-

ers belonging to a group are able to shift profits with-

in the group and thus, reduce tax payments (Chang

and Jeng 2015; Cole and McCullough 2006; Wang et

al. 2008).

It should be noted that commercial lines (Com_

line) are essentially more risky and contain higher

claim volatility than personal lines6. Based on the risk

taking viewpoint, insurers that write more commer-

cial lines also tend to purchase more reinsurance. In

addition, a higher claims volatility pattern also forces

insurers to rely more on reinsurance contracts (Ad-

ams, Hardwick, and Zou 2008; Graham and Rogers

2002; Graham and Smith 1999). Consequently, it is

expected that insurers that write greater volumes of

commercial lines are more likely to purchase rein-

surance. The commercial lines ratio is defined as: (di-

rect premium written of fire, allied lines, farm owners

multiple peril, commercial multiple peril, mortgage

guaranty, ocean marine, inland marine, financial guar-

anty, medical malpractice-occurrence, medical mal-

practice-claims made, earthquake, group accident and

health, credit accident and health (group and individ-

ual), other accident and health, workers’ compensa-

tion, other liability -occurrence, other liability-claims

made, products liability-occurrence, products liability-

claims made, commercial auto liability, aircraft (all

perils), fidelity, surety, burglary and theft, boiler and

machinery, and credit), divided by direct business

written.

6I sincerely thank the anonymous referee who proposed

that the underwriting portfolio has been considered to have

an effect on demand for reinsurance. Thus, this study

uses a commercial lines ratio to control underwriting mix

effects on the insurer’s demand for reinsurance.


Insolvency concerns also have an important in-

fluence on demand for reinsurance7. Chen, Hamwi, and

Hudson (2001) and Shiu (2011) point out that insur-

ers with a higher probability of insolvency tend to

purchase more reinsurance. The reason is that an in-

solvent insurer with higher levels of reinsurance is

better positioned to confront higher external financial

costs in the capital markets. This study adopts the

natural logarithm of the RBC ratio (LnRBC) to proxy

an insurer’s degree of insolvency. The RBC ratio is

defined as (Total Adjusted Capital *100)/(2*Author-

ized Control Level), where the items are collected

from the five-year historical annual statement from

the National Association of Insurance Commissioners

(NAIC).

The literature also suggests that the variation

effects among the different lines of business may in-

fluence the insurer’s reinsurance decisions, because

some lines of business may have particular effects on

their demand for reinsurance; for instance, liability

related lines, in which higher agency costs give the

insurer greater incentive to purchase more reinsure-

ance. Consequently, this article also uses premiums

written in each line of business (W(i))8 to control for

the impact of variations in lines of business on de-

mand for reinsurance (Chang and Jeng 2015; Cole

and McCullough 2006; Mayers and Smith 1990; Shiu

2011; Shortridge and Avila 2004; Wang et al. 2008).

Finally, year dummies are also included to control

time-series heterogeneous effects as a whole.

3. Methodology

Based on the logic of Cummins and Sommer (1996)

and Baranoff and Sager (2002; 2003), Shiu (2011) con-

cludes that the insurer’s leverage has endogenous

influence in determining the insurer’s demand for

reinsurance. Thus, insurers tend to have a target un-

derwriting risk as a result to determine their demand

for reinsurance and leverage simultaneously. Chang

and Jeng (2015) further propose that liquidity main-

tenance also produces an important endogenous effect

on demand for reinsurance. Likewise, for approach-

ing underwriting risk targets, insurers determine their

demand for reinsurance, liquidity, and leverage en-

dogenously and simultaneously9. According to the ar-

7I sincerely thank the anonymous referee for considering

the solvency concerns that affect the insurer’s reinsure-

ance decision. 8The variable is defined as the ratios of premiums writ-

ten in each line of business to premiums written in all 27

lines of business. To avoid a singular matrix in the re-

gression, Mayers and Smith (1990), Cole and McCul-

lough (2006), Wang et al. (2008), and Chang and Jeng

(2015) suggest the model should exclude the financial

guaranty, international, and reinsurance lines. 9Accordingly, this study further implements a Hausman

endogeneity test to confirm whether the insurer’s lev-

erage and liquidity are endogenous with its demand for

reinsurance. First, the explanatory variables of the lev-

erage equation are identified according to Cummins and

Sommer (1996), Baranoff and Sager (2002; 2003), Shiu

gument of Chang and Jeng (2015) and Shiu (2011),

the 2SQR paradigm is performed to correct the en-

dogeneity bias, by replacing the endogenous regres-

sors with their fitted values of the reduced-form equa-

tions in this study (Amemiya 1982; Chen and Portnoy

1996; Kim and Muller 2004; Powell 1983). The tradi-

tional 2SLS approach only captures the average ef-

fects of covariates on demand for reinsurance. How-

ever, the 2SQR approach not only explores poten-

tially differential effects across the reinsurance dis-

tribution but also provides further insightful informa-

tion on how the insurer’s demand for reinsurance is

determined. Therefore, this study intends to comple-

ment the literature gap in terms of the 2SQR approach.

The 2SQR estimation procedure is introduced

as follows. Consider a liner model describes as:

1 0 2 1 1 2Y Y Z , (1)

1202 ZY , (2)

where is the dependent variable of interest, is

the endogenous regressor, is an exogenous vector

of , is the exogenous vector of , is the un-

observed structural error term of , and is the

disturbance terms of . Let '21,,1 ZZZ be a vec-

tor of all exogenous variables. The endogeneity of

is due to dependence between and , conditional

on .

In the first stage, the reduced-form equations

are set as10

:

VZY 2 (3)

From equation (3), the fitted values could be gener-

ated, i.e., 2ˆ ˆY Z .

(2011), and Chang and Jeng (2015). These variables in-

clude demand for reinsurance, liquidity, firm size, busi-

ness and geographic Herfindahl indices, single firm dum-

my, intra-group Herfindahl Index, New York-licensed

dummy, marketing channel dummy, organizational form

dummy, return on capital, natural logarithm of the RBC

ratio, and year dummies. The explanatory variables that

are used in the liquidity equation are obtained from the

existing literature (Bruinshoofd and Kool 2002; Chang

and Tsai 2014; Chang and Jeng 2015; John 1993; Kim,

Mauer, and Sherman 1998; Opler et al. 1999; Shiu 2006).

These variables contain demand for reinsurance, lev-

erage, business and geographic Herfindahl indices, 2

years’ loss development, firm size, stock dummy, single

firm dummy, return spread, firm growth rate, cash flow

volatility, reserves of liability and property lines, claims

of liability and property lines, and year dummies. The

result of Hausman’s specification test (F value = 423,

rejecting the null hypothesis) indicates that the endog-

eneity influences of the insurer’s leverage and liquidity

do exist. 10Instrument variables for the reduced-form equations

consist of all exogenous, control, and year dummy vari-

ables of demand for reinsurance, liquidity, and leverage

equations.

1Z

1Y 2Z2Y

1Y

2Y

2Y

Z

1Y 2Y


Table 1 Summary Statistics of Numerical Variables Variables Minimum Mean Median Maximum Std Dev

Reins 0.0000 0.3878 0.3449 0.9632 0.2903

Leverage 0.0419 1.7788 1.1542 12.4190 2.0024

2_years_loss -0.0005 -0.0000a -0.0000b 0.0004 0.0001

Size 14.5017 18.3070 18.2111 22.9579 1.8366

Liq 0.2827 0.8284 0.8621 0.9927 0.1350

Tax_ex 0.0000 0.2899 0.2301 0.9566 0.2625

ROA -0.1631 0.0395 0.0408 0.2296 0.0592

Bus_H 0.1403 0.5796 0.5122 1.0000 0.3033

Geo_H 0.0420 0.5684 0.5272 1.0000 0.3857

Stock 0.0000 0.6781 1.0000 1.0000 0.4672

Single 0.0000 0.4093 0.0000 1.0000 0.4918

Com_line 0.0000 0.6310 0.8383 1.0000 0.3990

LnRBC 0.2425 2.1947 2.1403 4.7270 0.7673

Obs. 5,626

The dependent variable is Reins, which is defined as (affiliated reinsurance ceded + nonaffiliated reinsurance ceded)/

(direct business written plus reinsurance assumed). Leverage it is defined as the ratio of direct business written-to-

surplus. The variable of 2_years_loss is two-year loss development and equals the development of estimated losses

and loss expenses incurred two years before the current and prior year, scaled by policyholders’ surpluses. The proxy

of firm size is Size, and is defined as the natural logarithm of total assets. The Liq is defined as the sum of cash plus

invested assets-to-total assets. The variable of Tax_ex is defined as tax-exempt investment income relative to total

investment income. And ROA is defined as net income plus tax and interest expense, divided by total assets. The

Bus_H variable is the line of business Herfindahl Index, which is defined as the sum of the squares of the ratio of the

dollar amount of direct business written in a particular line of insurance to the dollar amount of direct business across

all 27 lines of insurance. Conversely, the Geo_H variable is the geographic Herfindahl Index, which is defined as the

sum of the squares of the ratio of the dollar amount of direct business in state j to the total amount of direct business

across all states. The organizational form dummy variable is Stock, which is to indicate stock or mutuals. It equals 1

if the insurer is a stock, and 0 if it is a mutual. The single dummy variable is Single, which is to indicate an affiliated

or nonaffiliated insure. It equals 1 if the insurer is nonaffiliated, and 0 if it is affiliated. Com_line represents the

proportion of the insurer’s commercial lines of business. Finally, LnRBC is defined as the natural logarithm of RBC

ratio. In this study, total firm-year observations are 5,626.

Note a: The mean value of 2_years_loss equals to −4.6591*10−5 rather than 0.

Note b: The median value of 2_years_loss equals to −3.2263*10−5 rather than 0.

In the second stage, in equation (1) is re-

placed by ; then, the 2SQR estimator of is

the solution to the following minimization problem:

(4)

To examine the distinct influences on deter-

minants of the insurer’s demand for reinsurance be-

tween higher and lower demand for reinsurance quan-

tiles, the equality test of estimated parameters of each

variable across quantiles will be implemented. In ad-

dition, diagnostic tests of model fitness are also con-

ducted to test whether the distribution of the distances

of demand for reinsurance is normal and whether the

quantile regression fits the data appropriately11

.

4. Data and Empirical Results

4.1 Data

An analysis is performed using data from pro-

perty-liability insurance companies in the NAIC for

the years 2006-2010, which originally comprised 3,007

11This study provides the Quantile-Quantile probability

plots and a histogram of the standardized residuals to

examine the model fitness issue.

insurers. To be included in the sample, each company

needed a complete data set for each single year. Thus,

insurers with missing data were deleted, and 2,060

insurers remained. Companies with unreasonable val-

ues (or illogical values) for all variables were also

excluded from the empirical analysis12

. Subsequently,

1,869 insurers remained. Insurers that operated as pro-

fessional reinsurers and whose reinsurance accounts

for more than 75% of their total premium written

were also excluded (Chang and Jeng 2015; Cole and

McCullough 2006; Powell and Sommer 2007; Shiu

2011). Finally, the exclusions left this study with com-

plete data for 1,773 insurers and 5,626 firm-year ob-

servations. In addition, to control for the outlier prob-

lem, the variables used in this study were winsorized

at the 1st and 99th percentiles, except for the dummy

variables, to avoid the influences of extreme values.

Summary statistics for all variables of the pool-

ed time-series and cross-sectional data are shown in

Table 113

. The mean value of reinsurance ratio (Reins)

is 38.78%, with 29.03% standard deviation. The aver-

age liquidity (Liq) is 82.84%, which proposes that in-

surers are conservative, and ranges from a minimum

12For instance, the reinsurance ratio <0 and >1, leverage

<0, the geographic and business Herfindahl Index >1

and/or <0. 13Table 1 does not report the numbers of other control

variables and year dummies.

1 0 2 1 21

1 0 2 1 1 2

1 0 2 1 1 2

1 0 2 1 1 21

K ˆR Y Y Z

ˆY Y Z

ˆMin Y Y Z

ˆ ( ) Y Y Z

2Y

2Y ˆ ( )

Tab

le 2

D

escri

pti

on

of

Each

Rein

su

ran

ce D

ecil

e

Des

crip

tion

of

ind

epen

den

t v

ari

ab

les

for

each

rei

nsu

ra

nce

dec

ile (

10

deci

les)

Qu

an

tile

(by

Rei

ns)

R

ein

s L

ever

ag

e 2

_ye

ars

_lo

ss

Siz

e L

iq

Ta

x_

ex

RO

A

Bu

s_H

G

eo_

H

Sto

ck

Sin

gle

C

om

_li

ne

Ln

RB

C

Dec

ile

1

0.0

04

13

7

1.1

18

06

0

-0.0

00

058

17

.24

56

66

0.8

76

40

3

0.2

52

01

8

0.0

45

07

2

0.8

02

13

5

0.7

12

64

1

0.6

72

59

8

0.6

92

17

1

0.6

12

62

6

2.1

74

68

9

Dec

ile

2

0.0

62

20

8

1.0

81

46

4

-0.0

00

060

18

.87

12

30

0.8

75

57

5

0.3

39

53

4

0.0

47

72

0

0.6

09

42

3

0.6

27

61

2

0.5

59

50

3

0.5

15

09

8

0.5

95

03

2

2.2

37

08

5

Dec

ile

3

0.1

28

99

2

1.0

66

56

0

-0.0

00

068

18

.20

30

86

0.8

78

65

2

0.2

88

59

0

0.0

44

93

7

0.5

92

17

2

0.6

48

27

0

0.4

84

90

2

0.6

07

46

0

0.6

83

69

4

2.2

56

22

6

Dec

ile

4

0.2

04

42

3

1.1

33

03

6

-0.0

00

054

18

.27

68

56

0.8

68

61

8

0.2

71

06

4

0.0

38

91

6

0.5

66

24

3

0.6

30

89

0

0.4

77

79

8

0.5

91

47

4

0.7

24

01

5

2.1

74

00

1

Dec

ile

5

0.2

96

46

3

1.0

57

48

2

-0.0

00

052

18

.35

01

09

0.8

53

70

8

0.2

79

92

5

0.0

40

73

2

0.5

63

62

0

0.5

89

53

0

0.6

02

13

1

0.4

81

35

0

0.7

14

53

7

2.1

18

12

7

Dec

ile

6

0.3

97

60

2

1.2

08

59

3

-0.0

00

054

18

.38

62

95

0.8

44

43

4

0.2

76

41

2

0.0

38

23

9

0.5

60

10

2

0.5

70

56

0

0.7

14

03

2

0.3

73

00

2

0.6

83

20

3

2.1

12

46

3

Dec

ile

7

0.5

0112

8

1.3

53

57

5

-0.0

00

042

18

.56

23

12

0.8

24

77

3

0.2

83

95

5

0.0

34

42

7

0.5

47

62

9

0.5

27

98

2

0.7

81

52

8

0.2

75

311

0.5

85

21

5

2.0

97

26

6

Dec

ile

8

0.6

18

94

8

1.9

67

29

6

-0.0

00

041

18

.62

94

98

0.7

78

79

8

0.3

14

13

1

0.0

33

53

6

0.5

16

05

3

0.4

80

24

8

0.8

20

60

4

0.2

18

47

2

0.5

53

05

7

2.0

81

01

5

Dec

ile

9

0.7

62

80

7

2.8

56

31

3

-0.0

00

024

18

.38

01

66

0.7

61

911

0.2

90

28

1

0.0

38

18

8

0.5

21

43

0

0.4

67

96

1

0.8

24

15

6

0.2

04

26

3

0.5

94

05

2

2.1

53

74

8

Dec

ile

10

0

.90

307

8

4.9

61

83

7

-0.0

00

014

18

.16

22

35

0.7

2110

7

0.3

03

21

9

0.0

33

55

5

0.5

17

64

3

0.4

27

94

8

0.8

44

64

3

0.1

33

92

9

0.5

64

13

2.5

44

63

4

This

tab

le p

rese

nts

the

mea

ns

of

each

rei

nsu

rance

dec

ile

for

all

var

iable

s. T

he

dec

ile

1 i

s th

e gro

up o

f th

e lo

wes

t R

ein

s gro

up,

wher

eas

dec

ile

10 i

s th

e hig

hes

t R

ein

s gro

up.

The

Rei

ns

is

def

ined

as

(aff

ilia

ted r

einsu

rance

ced

ed +

nonaf

fili

ated

rei

nsu

rance

ceded

)/(d

irec

t busi

nes

s w

ritt

en p

lus

rein

sura

nce

ass

um

ed).

Conce

rnin

g a

bout

Lev

erage,

it

is d

efin

ed a

s th

e ra

tio o

f dir

ect

busi

nes

s w

ritt

en-t

o-s

urp

lus.

The

var

iable

of

2_ye

ars

_lo

ss i

s tw

o-y

ear

loss

dev

elopm

ent

and e

qual

s th

e dev

elopm

ent

of

esti

mat

ed l

oss

es a

nd l

oss

expen

ses

incu

rred

tw

o y

ears

bef

ore

the

curr

ent

and p

rior

yea

r, s

cale

d b

y p

oli

cyhold

ers’

surp

luse

s. T

he

pro

xy

of

firm

siz

e is

Siz

e, a

nd i

t is

def

ined

as

nat

ura

l lo

gar

ithm

of

tota

l as

sets

. T

he

Liq

is

def

ined

as

the

sum

of

cash

plu

s in

ves

ted

asse

ts-t

o-t

ota

l as

sets

. T

he

var

iable

of

Tax_ex

is

def

ined

as

tax

-exem

pt

inves

tmen

t in

com

e re

lati

ve

to t

ota

l in

ves

tmen

t in

com

e.

And R

OA

is

def

ined

as

net

inco

me

plu

s ta

x a

nd i

nte

rest

expen

se,

div

ided

by

tota

l as

sets

. T

he

Bu

s_H

var

iable

is

the

line

of

busi

nes

s H

erfi

ndah

l In

dex

, w

hic

h i

s def

ined

as

the

sum

of

the

squar

es o

f th

e ra

tio o

f th

e doll

ar a

mount

of

dir

ect

busi

nes

s w

ritt

en i

n a

par

ticu

lar

line

of

insu

rance

to t

he

doll

ar a

mount

of

dir

ect

busi

nes

s ac

ross

all

27 l

ines

of

insu

rance

. C

onver

sely

, th

e G

eo_H

var

iab

le i

s th

e geo

gra

phic

Her

findah

l In

dex

, w

hic

h i

s def

ined

as

the

sum

of

the

squar

es o

f th

e ra

tio o

f th

e doll

ar a

moun

t of

dir

ect

busi

nes

s in

sta

te j

to t

he

tota

l am

ount

of

dir

ect

busi

nes

s ac

ross

all

sta

tes.

The

org

aniz

atio

nal

form

dum

my

var

iable

is

Sto

ck

, w

hic

h

is t

o i

ndic

ate

stock

or

mutu

als.

It

equal

s 1 i

f th

e in

sure

r is

a s

tock

, an

d 0

if

it i

s a

mutu

al.

The

single

dum

my

var

iable

is

Sin

gle

, w

hic

h i

s to

indic

ate

an a

ffil

iate

d o

r nonaf

fili

ated

insu

re.

It e

qual

s

1 i

f th

e in

sure

r is

nonaf

fili

ated

, an

d 0

if

it i

s af

fili

ated

. C

om

_li

ne

repre

sents

the

pro

port

ion o

f th

e in

sure

r’s

com

mer

cial

lin

es o

f busi

nes

s. F

inal

ly,

Ln

RB

C i

s def

ined

as

the

nat

ura

l lo

gar

ithm

of

RB

C r

atio

.



Figure 1 Distribution of Residuals for Reinsurance with 0.1Quantile

Figure 2 Distribution of Residuals for Reinsurance with 0.25 Quantile




Figure 6 Q-Q Plot of Residuals for Reinsurance with 0.1Quantile


Figure 7 Q-Q Plot of Residuals for Reinsurance with 0.25 Quantile




of 28.27% to a maximum of 99.27% of the total

assets. The mean value of leverage is approximately

1.7788 and ranges from 0.0419 to 12.4190. The mean

(median) of two-year loss development (2_years_loss)

is −4.6591*10−5

(−3.2263*10−5

), which indicates that

insurers tend to create over loss reserving. In addition,

Table 1 shows that 67.81% of observations are for

stock insurers and that 40.93% of these are listed as

single insurers in the database. Overall, the summary

statistics for the explanatory variables are similar to

those reported in previous studies (Chang and Jeng

2015; Cole and McCullough 2006; Fier, McCullough,

and Carson 2013; Garven and Lamm-Tennant 2003;

Mayers and Smith 1990; Wang et al. 2008) and indi-

cates that the sample selection in this study is appro-

priate.

Table 2 reports the mean values of explanatory

variables for 10 quantile deciles of the insurer’s rein-

surance distribution. Overall, before considering the

interactive impacts among explanatory variables, most

explanatory variables tend to have different patterns

in the lower and higher reinsurance quantiles. For in-

stance, as reinsurance quantiles increase, the mean

values of Liq, ROA, Bus_H, Geo_H, and Single de-

crease, whereas the averages of Tax_ex and 2_years

_loss increase. Interestingly, Leverage, Stock, and

LnRBC present a U-shape from the lower reinsurance

deciles to the higher reinsurance deciles, whereas an

inverse U-shape emerges for Size and Com_line.

These important features of higher and lower rein-

surance quantiles imply that analysis using the tradi-

tional 2SLS approach may be biased or insufficient.

Thus, this study proposes the 2SQR approach to re-

solve the inadequacies of the traditional 2SLS ap-

proach.

4.2 Empirical Results

Model fitness checks should be done before the

empirical analysis14

. This study implements two diag-

nostic tests of model fitness, a histogram of the stand-

ardized residuals and the quantile-quantile plot diag-

nostic to scrutinize whether the data is well-fitted or

not. Figure 1 to Figure 10 describe the model fitness

patterns for the standardized residuals and the quan-

tile-quantile plot diagnostic. Figure 1 to Figure 5 sug-

gest that higher quantile regressions ( = 0.5, 0.75,

and 0.9) fit the data well, whereas lower quantile re-

14I sincerely thank the anonymous referee for suggesting

this important issue prior to the empirical analysis.


gressions ( = 0.1 and 0.25) do not. In addition, Fig- ure 6 to Figure 10 also propose that the data for higher quantile demand for reinsurance ( = 0.5, 0.75, and 0.9) follow the line clearly, which indicates a normal distance distribution. However, lower demand for reinsurance quantiles ( = 0.1 and 0.25) deviate from normal in the tails, which presents short tails at both ends of the data distribution. In sum, model fitness checks propose that higher quantile regressions fit the data better than lower quantile regressions and present a normal distribution for reinsurance residuals.

To make a performance comparison between the 2SLS and 2SQR approaches, this study presents corresponding results from these two approaches in Table 3. The coefficient of Liq is significantly negative in the 2SLS model, which suggests that the sub- stitution hypothesis is supported (Chang and Jeng 2015; Hau 2006). However, the coefficients in the 2SQR model are significantly different in the higher and lower quantiles. For insurers in the lower reinsurance quantiles ( = 0.1, 0.25, and 0.5), liquidity presents negatively to demand for reinsurance. These results are consistent with the results obtained using the 2SLS model. Insurers with more liquid assets have a greater ability to mitigate unexpected and emergent demand for cash, which then enables them to decrease their demand for reinsurance, even though the reinsurance protection from reinsurers is lower. Con- versely, a significantly positive coefficient is found for insurers in the higher reinsurance quantile ( = 0.9). This result contradicts the hypothesis prediction that an insurer with a high level of reinsurance will need “less” reinsurance to uphold its target level of underwriting risk in the higher reinsurance quantiles. One plausible explanation for this interesting finding is that the higher liquidity insurer, with a higher given level of reinsurance, tends to have increased ability or capacity to underwrite a new or a risky business in advance and therefore, intends to increase its reinsurance purchasing as a whole.

In Table 3, the coefficient of the 2_years_loss is insignificantly negative in the 2SLS model. In addition, the results also present a different impact on demand for reinsurance between the 2SLS and 2SQR models. For insurers in the lower reinsurance group ( = 0.1, 0.25, and 0.5), the empirical results (negative coefficients) indicate that insurers with lower loss reserves tend to purchase less reinsurance, which is inconsistent to the expectation (Chang and Jeng 2015; Cole and McCullough 2006; Wang et al. 2008). This study speculates that insurers with lower levels of reinsurance tend to have higher liquidity, lower leverage, and higher-profit insurers

15 such that they

tend to purchase less reinsurance. Conversely, for insurers in the higher reinsurance group ( = 0.9), loss reserving is positively related to the demand for reinsurance, which indicates that a decrease in loss reserving will encourage insurers to purchase more reinsurance.

15Table 2 notes that insurers in the lower reinsurance

quantiles tend to have higher liquidity, lower leverage,

and higher profits. As a result, insurers are able to de-

crease their demand for reinsurance even though they

remain in a position of under-loss reserving.

Some independent variables are consistent to

the predictions for both the 2SLS and 2SQR models:

Reinsurance is negatively related to Tax_ex, ROA,

Bus_H, Geo_H, Single and Size, and is positively re-

lated to Leverage, Stock, Com_line, and LnRBC. In

Table 3, both the 2SLS and 2SQR models propose

that the tax-exempt factor is negatively related to the

insurer’s demand for reinsurance, which is inconsist-

ent with the suggestions of Smith and Stulz (1985),

Mayers and Smith (1990), and Garven and Lamm-

Tennant (2003). However, these results are closer to

the findings of Shiu (2011). To decrease expected tax

liabilities, an insurer with a higher marginal tax rate

will purchase less reinsurance. Furthermore, geograph-

ic and business diversified insurers tend to demand

more reinsurance than nongeographic and nonbusi-

ness diversified insurers. These results suggest that

insurers tend to rely on more real services via special-

ized knowledge and economies of scale from rein-

sureers. Moreover, consistent with Shiu (2011), more

concentrated insurers could specialize in, and may

underwrite, less volatile lines of business or less-risky

insured and therefore demand less reinsurance. In

addition, the results of the stock dummy variable

(Stock) indicate that the agency cost hypothesis is

supported for both the 2SLS and 2SQR models (Ad-

ams 1996; Shiu 2011). To mitigate the underinvest-

ment problem among residual claimholders, stock in-

surers purchase more reinsurance than mutual insurers.

As expected, the results displayed in Table 3

show that both the 2SLS and 2SQR models present a

negative relationship between the single dummy vari-

able (Single) and the demand for reinsurance. Single

insurers tend to demand less reinsurance because they

cannot benefit in terms of shifting profits within the

group and thereby reduce tax payments (Chang and

Jeng 2015; Cole and McCullough 2006; Shiu 2011;

Wang et al. 2008). Moreover, consistent with the argu-

ment of greater ability to confront losses and finan-

cial pressures, higher profitability insurers tend to de-

mand less reinsurance than lower profitability insur-

ers. In addition, consistent with the literature, this

study also finds that smaller insurers tend to confront

a higher probability of insolvency and, therefore, pur-

chase more reinsurance contracts than larger insurers.

In Table 3, the coefficients of Leverage are sig-

nificantly positively related to the demand for rein-

surance for both the 2SLS and 2SQR models. These

results are consistent with the hypotheses of bank-

ruptcy cost, agency cost, risk bearing, and renting

capital (Chang and Jeng 2015; Cole and McCullough

2006; Garven and Lamm-Tennant 2003; Shiu 2011;

Wang et al. 2008). In addition, the empirical results

indicate that the impacts of leverage on insurers’

demand for reinsurance differ between the lower and

higher quantiles. The effect of leverage on reinsure-

ance is likely to increase in higher levels of reinsure-

ance. Consequently, for each increment leverage, an

insurer with a high level of reinsurance will need

“more” reinsurance to maintain its underwriting risk

at a target level. In contrast, for each additional unit

of leverage, an insurer with a low level of reinsurance

will need “less” reinsurance in the lower quantiles.

Tab

le 3

E

mp

iric

al R

esu

lts

E

xp

ecte

d s

ign

2S

LS

2S

QR

/ Q

uan

tile

s (

=0.1

, 0.2

5, 0.5

, 0.7

5, an

d 0

.9)

0.1

0.2

5

0.5

0.7

5

0.9

Tes

t of

Eq

uali

ty

(X2 V

alu

e)

Inte

rcep

t

0.8

390

***

0.5

251

***

0.3

658

0.8

138

*

0.6

094

0.2

710

Leve

rage

+

0.0

789

**

-0.0

086

0.0

562

***

0.0

984

***

0.1

199

***

0.1

337

***

50.1

696

***

2_ye

ars

_lo

ss

+

-9.9

097

-2

4.7

115

**

-93.4

328

***

-80.1

537

**

75.3

432

140.5

779

**

49.6

809

***

Siz

e

- -0

.0128

***

-0.0

017

-0

.0008

-0

.0088

*

-0.0

27

2

***

-0.0

241

***

52.9

076

***

Liq

-

-0.6

086

***

-0.5

302

***

-0.5

163

*

-0.8

582

*

0.1

929

0.7

897

**

27.0

149

***

Tax_ex

+/-

-0

.0448

***

-0.0

006

-0

.0320

**

-0.0

685

***

-0.0

396

*

-0.0

386

**

20.4

048

***

RO

A

- -0

.2026

**

0.0

219

-0

.1339

*

-0.2

842

**

-0.6

461

***

-0.6

034

***

68.3

956

***

Bu

s_H

+

/-

-0.1

050

***

-0.0

920

***

-0.1

137

***

-0.1

247

***

-0.0

324

0.0

126

46.7

855

***

Geo

_H

+

/-

-0.0

652

***

-0.0

086

-0

.0313

**

-0.0

428

**

-0.0

982

***

-0.0

856

***

38.6

923

***

Sto

ck

+/-

0.0

717

***

0.0

410

***

0.0

667

***

0.0

836

***

0.1

019

***

0.0

584

***

26.0

646

***

Sin

gle

-

-0.1

458

***

-0.0

140

***

-0.0

779

***

-0.1

869

***

-0.2

292

***

-0.1

917

***

378.7

580

***

Com

_li

ne

+

0.0

779

***

0.0

752

***

0.1

046

***

0.0

787

**

0.0

911

***

0.0

736

**

3.7

495

Ln

RB

C

+

0.1

001

***

0.0

094

*

0.0

665

***

0.1

367

***

0.1

171

***

0.0

870

***

119.5

736

***

Obs.

5,6

26

This

tab

le p

rese

nts

the

resu

lts

of

both

the

2S

LS

appro

ach (

colu

mn 3

) an

d t

he

2S

QR

appro

ach w

ith q

uan

tile

s =

0.1

, 0.2

5,

0.5

, 0.7

5, an

d 0

.9 (

colu

mn 4

~ 8

). C

olu

mn 2

rep

ort

s th

e pre

dic

tion s

igns

of

firm

-spec

ific

char

acte

rist

ics.

Colu

mn 9

show

s th

e te

st o

f co

effi

cien

ts e

qual

ity

for

each

var

iable

acr

oss

quat

nle

s. T

he

dep

enden

t var

iable

is

Rei

ns,

whic

h i

s def

ined

as

(aff

ilia

ted r

einsu

rance

ceded

+

nonaf

fili

ated

re

insu

rance

ce

ded

)/(d

irec

t busi

nes

s w

ritt

en plu

s re

insu

rance

as

sum

ed).

L

eve

rage

it is

def

ined

as

th

e ra

tio of

dir

ect

busi

nes

s w

ritt

en-t

o-s

urp

lus.

T

he

var

iable

of

2_ye

ars

_lo

ss i

s tw

o-y

ear

loss

dev

elopm

ent

and e

qual

s th

e dev

elopm

ent

of

esti

mat

ed l

oss

es a

nd l

oss

expen

ses

incu

rred

tw

o y

ears

bef

ore

the

curr

ent

and p

rior

yea

r, s

cale

d b

y p

oli

cyhold

ers’

surp

luse

s. T

he

pro

xy

of

firm

siz

e is

Siz

e,

and i

s def

ined

as

the

nat

ura

l lo

gar

ithm

of

tota

l as

sets

. T

he

Liq

is

def

ined

as

the

sum

of

cash

plu

s in

ves

ted a

sset

s-to

-tota

l as

sets

. T

he

var

iable

of

Tax_ex

is d

efin

ed a

s ta

x-e

xem

pt

inves

tmen

t in

com

e re

lati

ve

to t

ota

l in

ves

tmen

t in

com

e. A

nd R

OA

is

def

ined

as

net

inco

me

plu

s ta

x a

nd i

nte

rest

expen

se,

div

ided

by

tota

l as

sets

. T

he

Bu

s_H

var

iable

is

the

line

of

busi

nes

s H

erfi

ndah

l In

dex

, w

hic

h i

s def

ined

as

the

sum

of

the

squar

es o

f th

e ra

tio o

f th

e doll

ar a

mount

of

dir

ect

busi

nes

s w

ritt

en i

n a

par

ticu

lar

line

of

insu

rance

to t

he

doll

ar a

mount

of

dir

ect

busi

nes

s ac

ross

all

27 l

ines

of

insu

rance

. C

onver

sely

, th

e G

eo_H

var

iable

is

the

geo

gra

phic

Her

findah

l In

dex

, w

hic

h i

s def

ined

as

the

sum

of

the

squar

es o

f th

e ra

tio o

f th

e doll

ar

amount

of

dir

ect

busi

nes

s in

sta

te j

to t

he

tota

l am

ount

of

dir

ect

busi

nes

s ac

ross

all

sta

tes.

The

org

aniz

atio

nal

form

dum

my

vari

able

is

Sto

ck

, w

hic

h i

s to

indic

ate

stock

or

mutu

als.

It

equal

s 1 i

f

the

insu

rer

is a

sto

ck,

and 0

if

it i

s a

mutu

al.

The

single

dum

my

var

iable

is

Sin

gle

, w

hic

h i

s to

indic

ate

an a

ffil

iate

d o

r nonaf

fili

ated

insu

re.

It e

qual

s 1 i

f th

e in

sure

r is

nonaf

fili

ated

, an

d 0

if

it i

s

affi

liat

ed.

Com

_li

ne r

epre

sents

the

pro

port

ion o

f th

e in

sure

r’s

com

mer

cial

lin

es o

f busi

nes

s. F

inal

ly,

Ln

RB

C i

s def

ined

as

the

nat

ura

l lo

gar

ithm

of

RB

C r

atio

. In

this

stu

dy,

tota

l fi

rm-y

ear

obse

rvat

ions

are

5,6

26.

***, **, an

d *

rep

rese

nt

stat

isti

cal

signif

ican

ce a

t th

e 1%

, 5%

, an

d 1

0%

lev

els,

res

pec

tivel

y.



Furthermore, consistent with the risk taking

viewpoint and higher claims volatility argument, the

results indicate that insurers that write more comer-

cial lines (Com_line) tend to purchase more reinsure-

ance (Adams, Hardwick, and Zou 2008; Graham and

Rogers 2002; Graham and Smith 1999). Finally, both

the 2SLS and 2SQR models suggest that insurers

with higher probability of insolvency tend to pur-

chase more reinsurance, which is consistent with the

argument of insolvency concerns (Chen, Hamwi, and

Hudson 2001; Shiu 2011).

Note that insurers in the higher reinsurance

quantiles are safer than those in the lower reinsurance

quantiles16

. Thus, it is predicted that the determinants

on demand for reinsurance in the lower and higher

quantiles are distinct. In other words, the influence of

the demand for reinsurance for each variable may

present a different sign, or a dissimilar impact (mag-

nitude). This study further implements the tests of e-

quality for each variable across the five quantiles ( =

0.1, 0.25, 0.5, 0.75, and 0.9). Testing these results

indicates that there is a difference among the esti-

mated parameters for all variables across the five

quantiles, except Com_line, which indicates that Hy-

pothesis 1 is supported. Specifically, Leverage, ROA,

Geo_H, Single, and Size have larger impacts on de-

mand for reinsurance in the higher reinsurance quan-

tiles, whereas Bus_H presents an inverse result in the

lower reinsurance quantiles. In addition, the evidence

of Tax_ex, Stock, and LnRBC suggests that the influ-

ence on demand for reinsurance is larger in the me-

dian quantile. This study also finds that the signs of

Liq and 2_years_loss are significantly different be-

tween the lower and higher reinsurance quantiles.

However, the other remaining variables present a con-

sistent sign regarding both lower and higher reinsure-

ance quantiles. Summing up, the evidence suggests

that Hypothesis 2 is, as a whole, weakly supported.

Taken together with previous empirical results,

the evidence presented by this study proposes that the

2SQR approach reports a definite and distinct impact

on the demand for reinsurance for various reinsurance

quantiles. As proposed, the traditional 2SLS approach

was found to contain some inappropriate assumptions

and therefore to produce a biased estimation. In con-

trast, the 2SQR approach provides efficient estima-

tion and insightful information on the determinants of

demand for reinsurance, which could compensate for

16I sincerely thank the anonymous referee who pointed

out that if insurers try to maintain a target level of under-

writing risk, then the argument that “insurers in the high-

er reinsurance quantiles are safer than those at the lower

reinsurance quantiles” may not hold theoretically. In

general, it should be noted that the protections from rein-

surance contracts are higher for insurers in the higher re-

insurance quantiles. Insurers purchase more reinsurance

could mitigate not only underwriting risk but also liquid-

ity risk (or short-term financial pressure), catastrophic

risk, and bankruptcy risk. Even if insurers tend to main-

tain a target level of underwriting risk, they may still

have incentives to demand more reinsurance as a whole.

the inadequacy of the traditional 2SLS approach. Over-

all, this study concludes that Hypothesis 3 is strongly

supported.

5. Concluding Remarks

Instead of using the traditional 2SLS approach, this

study proposes the 2SQR approach for examining the

determinants of the insurer’s demand for reinsurance.

This study finds that the determinants of demand for

reinsurance are driven by the insurer’s specific quan-

tile within the overall demand for reinsurance distri-

bution. The empirical evidence shows that the param-

eters (signs) for liquidity and loss development vary

within different reinsurance quantiles, which differen-

tiates these findings from previous ones (Chang and

Jeng 2015; Cole and McCullough 2006; Hau 2006;

Shiu 2011; Wang et al. 2008). The evidence indicates

that the traditional 2SLS approach may provide an

insufficient or biased explanation for the determinants

of insurers’ demand for reinsurance. In addition, al-

though consistent evidence is presented for other

firm-specific characteristics, the magnitudes of these

effects on demand for reinsurance differ significantly

across all reinsurance quantiles. Again, these new find-

ings, which also differ from the literature, enrich the

function of the 2SQR analysis in the lower and higher

quantiles of the reinsurance distribution. To sum up,

the 2SQR approach is more efficient and can provide

more insightful information than the traditional 2SLS

approach.

Reinsurance is an important issue for the pri-

mary insurers. Policyholders, policy makers, and/or

regulators should pay more attention in their assess-

ments of an insurer’s financial pressures and bank-

ruptcy problems, especially for insurers operating with

lower levels of reinsurance. In sum, it is of impor-

tance for policyholders, policy makers, and/or regula-

tors to assess the insurer’s demand for reinsurance by

considering the empirical implications of both the

2SLS and 2SQR models.


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