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Underwriting profits and capital market: a simultaneous equation approach.


Underwriting profits in the US property and casualty insurance industry have not been stable for the past several decades. The cyclicality of underwriting results is referred to as underwriting cycle. There can be many causes of the volatility of underwriting results (Corning & Company, 1979). It is known that insurers sell insurance policies in the insurance market and receive premiums in advance. Since losses can occur later throughout the year, they can invest premium reserves in the capital market. The source of investment income is the time lag between the collection of premiums and the payment of claims. For many cases, they are willing to sell policies at lower premiums as long as investment returns or bond yields are attractive. Bonds are their major investments (Best's Aggregates & Averages, 2013).

This cash flow underwriting practice can lead to underwriting losses, and insurers are willing to accept the down cycle of the profit as long as they expect sufficient investment income to offset underwriting losses. However, when interest rates or bond yields are low, insurers have to depend upon insurance business and raise premiums, resulting in higher underwriting profits of the up cycle. The cycle can take a few years because insurers cannot adjust to new business conditions instantaneously (Doherty & Kang, 1988). They must obtain a regulatory approval in most states and wait until policies are expired for a rate change. Therefore, it is difficult to explain the unstable underwriting returns without considering capital market conditions.

Historically speaking, property and casualty insurers in the U.S have been heavily dependent upon investment income and have experienced underwriting losses for many years. As shown in Table 1, the average underwriting loss was more than 15 billion dollars during the 1992 to 2012 period, whereas all US insurers were able to earn an average investment income of about 44 billion dollars during the same period (Best's Aggregates & Averages-Property/Casualty, 2013). Furthermore, Figure 1 shows that insurers were able to make underwriting gains for only 3 years (2004, 2006 & 2007) during the 20 year period. This figure suggests that property-liability insurers were willing to sacrifice underwriting profits to obtain cash flows when investment earnings are expected to be high. To the extent that negative or very small amount of underwriting profits combined with investment earnings are sufficiently enough to satisfy stockholders, insurers expand premiums written and pursues their goal of premium growth. The US insurance market is very competitive, and a rate and service competition cannot be avoided.

Previous studies on the subject tried to discover the main causes of the cyclical profits within the framework of the insurance market alone. This study attempts to explain fluctuating underwriting profits based on both insurance and capital market equilibriums. This study postulates that equilibrium profits may not be stable over time and that insurers collectively adjust toward a changing equilibrium in a competitive market. An equilibrium rate of return on underwriting can be derived from the capital asset pricing model. In this analysis, the fluctuation of the profit margin is viewed as a movement toward equilibrium due to the combined competitive forces of insurance and capital markets. This study presents a simultaneous equation model of the joint equilibrium of insurance and capital markets. It may have an implication not only for insurers to make competitive premium rates, but for regulations to impose a proper rate regulation in a timely manner.


There have been many studies of the underwriting profit cycle that tried to determine the causes of the cycle. The classic study by Stewart (1981) explained that the profit cycle existed because of price competition among insurers and the flexibility of insurance supply. The author emphasized the impact of external environment changes and rate regulation on the profit cycle. The Stewart study also identified the various patterns of the profit cycle among insurance lines. On the other hand, the Conning and Company (1979) study considered the underwriting cycle as the swing caused by a disparity between premium growth rates and changes in insurance costs. According to the study, the cycle existed because of an insurer's inability to price correctly and competitive pressures on the level of premium rates. The Conning study indicated that underwriting results were less cyclical in those states where insurers were allowed to price without regulatory interference than in those states where rates were regulated.

There were many other studies of the profit cycle that have sought to explain the causes of the cyclical profits within the framework of the insurance market (Berger, 1988; Cummins & Outreville, 1987; Venezian, 1985). They addressed the importance of investment earnings in relation to the profit cycle. The term cycle requires a proper statistical testing. Therefore, the study by Simmons and Cross (1986) used a cycle regression algorithm method to test if a cycle existed, and found a six-year cycle in the US property/liability insurance market. The study by Doherty and Kang (1988) was based on a joint equilibrium model of insurance demand and supply. Their findings suggested that the equilibrium insurance price was jointly determined by insurance and capital market conditions.

Other previous studies recognized that insurance capacity or the supply of capital was a major factor contributing to the cyclical nature of underwriting profits. The study by Higgins (2000) argues that the insurance cycle can exist when insurance capacity is limited, meaning that the cycle is caused by the supply or capacity restrictions. It implies that no cycle can exist if insurance companies can adjust to changing market equilibrium conditions with no restrictions. However, when insurers are subject to capacity restraint problems, they cannot adjust to new market conditions instantaneously, producing interruption and cyclical business patterns.

The study by Gron (1994) has a strong argument on the capacity-constraint theory and found a negative relationship between underwriting returns and capacity (i.e. a reduction in capacity causes higher insurance prices and improved profitability). In other words, when insurance companies face capacity or supply limitation, they cannot write a large volume of new policies. Under this situation, they have to raise insurance price and underwriting profits can be increased. On the other hand, when a plenty of reserves are available and insurance capacity is not restricted, insurers are willing to write more new policies. Under this situation, they have to cut insurance pricee, causing underwriting losses. The Gron's empirical study found the insurance capacity had a significant negative relationship with underwriting returns.

Another recent study (Kang & Domingo, 2012) on the capacity constraint indicated that the insurance supply was limited by the insurance capacity. The study offered a regression model and tried to explain the cycle in relation to interest rates and insurance capacity. Another study by Jawadi, Bruneau & Sghaier (2009) tested insurers' dependency to financial markets in five countries (Canada, France, Japan, the United Kingdom, and the United States) and found that a significant relationship between insurance business and financial markets. They analyzed the insurance cycle in the framework of nonlinear model of non-life insurance premium, interest rates and stock prices. The empirical study found a strong evidence of significant linkages between insurance and financial markets.


Sample and Data Collection

The study employs a 1951-2012 time series data of combined ratios, interest rates (3-month Treasury bill yields) and policyholders' surplus for all US property-liability firms. The interest data are collected from the Economic Report of the President Transmitted to the Congress (February, 2013) and all other insurance data are collected from the Best's Aggregates & Averages published by A.M. Best Company.

Equilibrium Underwriting Profit Model

Following the Biger and Kahane (1978) study an equilibrium underwriting profit model is as follows, assuming zero underwriting profit.

[R.sub.u] = - b [R.sub.f]


[R.sub.u] = "fair" rate of return on underwriting.

b = fund-generating coefficient.

[R.sub.f] = risk-free interest rate. (1)

As shown in the model, an equilibrium rate of return on underwriting is negatively related to the risk-free rate multiplied by the fund-generating coefficient (b). It implies that insurers are expected to incur underwriting losses in a competitive market. How much underwriting losses can be sustained depends upon the fund-generating coefficient and interest rates. The coefficient is a measure of investable funds generated per dollar of premiums written. An approximate measurement of the coefficient can be obtained by dividing the sum of premium reserves and loss reserves by earned premiums (Kang, 2012).

Simultaneous Equation

This study presents a regression model of simultaneous equations to explain the underwriting profit cycle in relation to interest rates, policyholders' surplus and investment earnings.

[([R.sub.u]).sub.t] = [B.sub.o] + [B.sub.i] [(INT).sub.t-1] + [B.sub.2] [(PHS).sub.t-1] (1)

[PHS.sub.t-1] = [B.sub.o] + [B.sub.i] [(IE).sub.t-1] + [B.sub.2] [(Ru).sub.t], where (2)

[([R.sub.u]).sub.t] = equilibrium rate of return on underwriting

[(INT).sub.t-1] = a lag of interest rates

[(PHS).sub.t-1] = a lag of policyholders' surplus

[(IE).sub.t-1] = a lag of investment earnings

The dependent variable in equation (i) is the underwriting profitability while the independent variables include interest rates and policyholders' surplus. When interest rates are high, insurers expect high investment earnings and are likely to sell more policies at a lower price, resulting in underwriting losses, Therefore, we expect a negative sign on the interest variable. The policyholder's surplus variable is included in the model to measure insurance capacity. Please note that how much insurance policies that an insurer can write depends on the available amount of policyholders' surplus. It means that an insurer with a large amount of surplus can sell more polices at lower premium rate, resulting in underwriting losses. Therefore, this study hypothesizes a negative relationship between policyholders' surplus and underwriting profits. Equation (2) indicates that policyholders' surplus in turn depends upon investment earnings and underwriting profits. These earnings and profits can be an addition to policyholders' surplus, so we expect a positive sign on the two earnings variables in equation (2). Please note that the simultaneous equation model is based on the interaction between investment earnings and underwriting profits.

A lag relation is considered because insurers cannot adjust instantaneously to new interest rate environments due to regulatory, accounting and policy period lags. (Kang, 2012) For example, in most states insurers must acquire a regulatory approval to use new premium rates. Also, they have to wait until policies are expired before new rates are implemented. For example, this year's interest rates and policyholders' surplus have an impact on current year's underwriting business that can result in the following year's underwriting profitability. Underwriting profitability is measured by the combined loss and expense ratios. Underwriting profits are one minus the sum of loss and expense ratios. The interest rate variable ([R.sub.f]) is measured by the yield on 3-month treasury bills.

Before attempting to estimate the model, it is important to determine whether its coefficients can be identified. An equation is unidentifiable if it is impossible to find the values of the parameters of a structural model, given the parameter values of a reduced form. In our model, endogenous variables are underwriting profit [([R.sub.u]).sub.t] and policyholders' surplus [(PHS).sub.t-1], while exogenous variables are interest rate variable [(INT).sub.t-1] and investment earning variable [(IE).sub.t-1]. Equation (1) is exactly identified because it has two endogenous variables and one excluded predetermined variable, [(IE).sub.t-1]. Equation (2) is also exactly identifiable because it has two endogenous variables and one excluded predetermined variable, [(INT).sub.t-1]. Therefore, the estimation of the model does not present any problem as far as identifications are concerned.

The two stage least square (2SLS) method is used for estimation. The application of the ordinary least square method does not yield consistent parameter estimates due to correlations between disturbance terms. The two stage least square (2SLS) method can yield consistent estimates by replacing the policyholders' surplus variable (an endogenous independent variable) for the estimated surplus variable. In the first stage, the surplus variable is regressed on the entire set of predetermined variables in the model. This is equivalent to estimating a reduced form. In the second stage, the underwriting profit variable, [([R.sub.u]).sub.t] is regressed on the "estimated" surplus variable and another exogenous variable in the model.


Both linear and log-linear models were estimated since the authors do not know an exact functional form of the simultaneous equation model. Table 2 reports the two-stage least square (2SLS) estimated results of a linear model whereas Table 3 includes the estimated results of a log-linear model. As shown in Table 2, the equation (1) has a R square of about 46% and a high F value = 23.39, meaning that the first equation of the linear simultaneous equation model is highly significant. Both independent variables in the first equation are hypothesized to have negative signs. The lag of the interest rate variable (LINT3) has an expected sign and is very significant, indicating a negative relationship between the interest rate variable and underwriting returns. The lag of the policyholders' variable (LPHS) in Equation (1) is also highly significant and carries an expected negative sign. It means that an insurer with a large amount of surplus sells more policies at lower premium rates, resulting underwriting losses. The second equation of the linear model has a very high R square of 93% with a high F value=375, indicating a highly significance of equation (2). Both the lag of investment earning variable (LIE) and the underwriting profit variable (UP) have positive signs and are very significant as hypothesized.

Table 3 reports the 2SLS estimated results of a log-linear model. The first equation's R square and F values are much lower than the linear model. Both independent variables (LINT3 and LPHS) in Equation (1) carry expected negative signs, but none of them are significant. The second equation's estimated results are similar to Equation (1) in the non-linear model. It has much lower R square and F values than the linear model. The investment income variable (LIE) in Equation (2) has a positive expected sign and is significant at the ninety-five percent level. The underwriting profit variable (UP) has the expected sign, but is not significant. Overall, it implies that the log-linear model estimation does not provide a proper fit of the model.


This study attempts to explain the cyclical underwriting business practices based on capital market conditions. Historical data strongly supports our view that the U.S. property and casualty insurers heavily depend upon investment earning and are willing to sacrifice underwriting profits. The capital asset pricing model can be applied to underwriting business and suggests that insurers can incur underwriting losses in equilibrium. Long-tail liability insurance lines are supposed to have larger underwriting losses than short-lag property lines. A simultaneous equation model is developed and estimated using the two-least square method (2SLS) based on interaction between insurance and capital market conditions. Both linear and log-linear models are estimated. The linear model's estimated results confirm these hypothesis that underwriting results are negatively related to interest rates and policyholder' surplus. The linear model seems to be a better fit for estimating the model.


Best's Aggregates & Averages, Property-Casualty Edition, A.M. Best Company, 2013.

Biger, N., & Yehuda, K. (1978). Risk Consideration in Insurance Ratemaking. The Journal of Risk and Insurance, 45, 121 - 132.

Conning & Company. (1979). A Study of Why Underwriting Cycles Occur, Insurance Management Services, Conning and Company.

Cummins, D. J., & Francois, O. J. (1987). An International Analysis of Underwriting Cycles in Property-Liability Insurance. The Journal of Risk and Insurance, 54 (2), 246 - 262.

Doherty, N. A., & Han, B. K. (1988). Interest Rates and Insurance Price Cycles. The Journal of Banking and Finance, 12, 199 - 214.

Economic Report of the President Transmitted to the Congress. (2013). US Government Printing Office, Washington, 2013.

Fairley, W. B. (1979). Investment Income and Profit Margins in Property-Liability Insurance: Theory and Empirical Results. Bell Journal of Economics. 10, 192-210.

Gron, A. (1994). Capacity Constraints and Cycles in Property--Casualty Insurance Markets. RAND. Journal of Economics, 25 (1), 110-127.

Higgins, M. L., & Thistle, P.D. (2000). Capacity Constraints and the Dynamics of Underwriting Profits. Economic Inquiry, 38 (3), 442.

Hill, R. D. (1979). Profit Regulation in Property-Liability Insurance. Bell Journal of Economics. 10, 172 - 191.

Jawadi, F., Bruneau, C., & Sghaier, N. (2009). Nonlinear Cointegration Relationships Between

Non-Life Insurance Premiums and Financial Markets. Journal of Risk and Insurance, 76 (3), 753-783.

Kahane, Y. (1978) .Generation of Investable Funds and the Portfolio Behavior of the Non- Life Insurers. The Journal of Risk and Insurance, 45 (1), 65 - 77.

Kang, H. B. & Domingo Joaquin. (2012). Interest Rates, Insurance Capacity and Cyclical Underwriting Profits Revisited," International Journal of Business, Accounting, and Finance, 6 (1), 132-141.

Stewart, B. D. (1981). Profit Cycles in Property-Liability Insurance.Issues in Insurance, CPCU 10, American Institute for Property and Liability Underwriters, 2, 2nd Edition.

Simmons, L. F., & Mark L. Cross. (1986). The Underwriting Cycle and the Risk Manager. Journal of Risk and Insurance, 53 (1), 155-163.

Venezian, E. C. (1985). Ratemaking Methods and Profit Cycles in Property and Liability Insurance. The Journal of Risk and Insurance, 52 (3), 477-500.

Han B. Kang

Illinois State University

Han B. Kang is a professor at the Department of Finance, Insurance and Law, Illinois State University, Normal, Illinois. He joined the University in 1983 and taught corporate finance, insurance and real estate courses. He has published in numerous scholarly journals including Journal of Banking and Finance, CPCU Journal, Journal of Insurance issues, Managerial Finance, Journal of Business Case Studies, Midwest Review of Finance and Insurance, Southern business Review, Journal of International Insurance, Journal of Real Estate Research, Journal of American Real Estate and Urban Economic Association, and International Journal of Business, Accounting and Finance among many other journals.

Table 1
Underwriting Gains/Losses and Investment Income (1992-2012)
(In Milions)

Year      Underwriting Gains/Losses   Net Investment Income

1992               -36,260                   33,734
1993               -18,094                   32,645
1994               -22,083                   33,687
1995               -17,375                   36,834
1996               -17,162                   37,962
1997               -6,030                    41,499
1998               -17,669                   41,097
1999               -24,750                   40,071
2000               -32,145                   42,650
2001               -52,692                   39,849
2002               -32,347                   41,099
2003               -5,230                    41,147
2004                1,692                    41,776
2005               -6,676                    51,879
2006               34,141                    54,826
2007               18,779                    58,054
2008               -21,809                   54,412
2009                -195                     50,912
2010               -10,365                   49,855
2011               -35,201                   51,356
2012               -13,806                   49,237
Average            -15,013                   44,028

Source: Best's Aggregates & Averages, 2013

Table 2
2SLS Estimated Results for a Linear Model

Equation (1)             Unstandardized

                        B        Std. Error

(Constant)              7.710         1.495

LINT3                  -1.432         0.230
LPHS                -1.432E-5         0.000
R Square = 0.455
F Value = 23.393

Equation (2)

(Constant)           -30351.2        9920.5

LIE UP                 11.181         0.448
R Square = 0.931       9136.3        1971.1
F Value = 375.2

Equation (1)        Standardized

                        Beta          t      Sig.

(Constant)                          5.157    0.000

LINT3                  0.661       -6.239    0.000
LPHS                   0.467       -4.252    0.000
R Square = 0.455
F Value = 23.393

Equation (2)

(Constant)                         -3.059    0.003

LIE UP                 1.060       24.986    0.000
R Square = 0.931       0.280        4.635    0.000
F Value = 375.2

Table 3
2SLS Estimated Results for a Non-Linear Model

Equation (1)           Unstandardized

                       B      Std. Error

(Constant)           2.830         1.853

LINT3               -0.301         0.464

LPHS                -0.168         0.179
R Square = 0.058
F Value = 0.737

Equation (2)

(Constant)           1.282         4.423

LIE                  1.051         0.383
UP                   1.065         2.030
R Square = 0.523
F Value = 13.182

Equation (1)        Standardized

                        Beta          t       Sig.

(Constant)                          1.527     0.140

LINT3                  -0.129      -0.649     0.523

LPHS                   -0.189      -0.942     0.355
R Square = 0.058
F Value = 0.737

Equation (2)

(Constant)                          0.290     0.774

LIE                     1.191       2.743     0.011
UP                      0.949       0.525     0.605
R Square = 0.523
F Value = 13.182
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Author:Kang, Han B.
Publication:International Journal of Business, Accounting and Finance (IJBAF)
Article Type:Report
Geographic Code:1USA
Date:Sep 22, 2014
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