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Interest rates and profit cycles: a disaggregated approach.

Interest Rates and Profit Cycles: A Disaggregated Approach


Changes in the performance of property-liability insurers have been dramatic, resulting in problems that affect the supply of insurance and the solvency ofinsurers. Numerous researchers have examined the behavior of returns in this industry and have concluded that they may be characterized as cyclical. This paper redefines the nature of returns that are studied and concludes that disaggregated models with interest rate terms perform better than simple autoregressive models in explaining the behavior of profits.


Recent movements in prices and profits in the property-liability insurance industry have led to the assertion that the industry returns are cyclical, and all previous empirical work supports this conclusion. Cycles in the underwriting profits of the property-liability industry have been attributed to a number of causes. Doherty and Kang (1988) attributed cycles to fluctuations in interest rates, Smith (1981) attributed them to regulatory lags, Stewart (1981) attributed them to changes in capital flows into the industry, Jablonski (1985) attributed them to adaptation under imperfect information, and Venezian (1985) and Cummins and Outreville (1987) attributed them to procedural lags in the process of estimating marginal costs.

Most prior work has examined the available data in the context of highly restrictive models. This paper seeks to expand on prior work in several ways. One is to introduce financial variables to reflect the effect of investment returns for individual lines of insurance. The second is to improve upon the by-line models of Venezian (1985) by allowing the error terms in various lines of insurance to be contemporaneously correlated. The third is to determine whether an aggregate model for the industry or separate models for individual lines are more consistent with the data.

Financial Variables

Models of insurance price determination [Biger and Kahane (1978), Kahane (1978), Fairley (1979), Hill (1979), Myers and Cohn (1981), Kraus and Ross (1982) and Venezian (1983)] lead to the conclusion that calculation of the required rate of return on equity from entering the insurance business should reflect both underwriting income and income derived from the investment of reserves and net worth. It is assumed that the measures of profit, both Joseph A. Fields is Assistant Professor of Finance at the University of Connecticut. Emilio C. Venezian is Chairman and Associate Professor of Business Administration at Rutgers University and President of Venezian Associates. underwriting and operating, are determined from a portfolio based model, where the level of return is based on the decision concerning the level of risk undertaken and exogenous economic variables.

In view of the general agreement that overall operating profits and returns depend on both underwriting and investment income, it is surprising that cycles have usually been analyzed using data on the underwriting rather operating profit.(1) Foster (1977) showed that underwriting ratios do not contain as much information as operating ratios concerning the economic value of the insurance company. The use of operating margins would lead to an analysis of the cyclical nature of the "true price" of insurance representing both the economic profit to the insurer and the opportunity cost of funds to the insured.

Interest Rate Effects

Most of the models that address rate of return from insurance essentially focus on the price that would occur in long-run equilibrium. Krauss and Ross (1982) argue that, in the context of their model, the risk free interest rate should be considered as a "real" or inflation adjusted rate if expected claim costs are set at current prices. On the other hand, if costs are based on forecasts that include inflation the appropriate rate of interest should be the nominal rate. In either event, unanticipated changes in purchasing power can have the effect of creating a difference between the intended and realized margins.

While the models of insurance price determination assume that the effect of interest rate changes on profit margins is Fisherian, Geske and Roll (1983) argue that evidence from the stock market does not support a Fisherian reaction to inflation. Changes in interest rates are likely to affect economic activity and therefore, real returns. The model of the cycle proposed by Doherty and Kang (1988) appeals to some of the same considerations. In their model the insurer attempts to set insurance prices by forecasting the value of short term interest rates and fluctuations in the experienced profit margin are related to errors in the forecast.

The empirical structure that developed here introduces the "unanticipated" interest rate (i.e., the difference between the short term rate and the value predicted by using a first-order expectations model) as an independent variable. The use of an autoregressive model to predict interest rates follows Nelson and Schwert (1977), who find that such models do a reasonable job of estimating future interest rates.

Prediction of the sign and magnitude of the unanticipated interest rate coefficient is not simple. Several factors, such as flexible product price and quality, Fisherian changes in required rates, and reserving errors would suggest that the coefficient should be positive and proportional to the amount of float in a line.2 The inability of the firm to adjust prices and/or service during periods of increasing interest rates would reduce the value of the coefficient. Under CAPM assumptions: E(r sub e) = r sub f + B(r sub m - r sub f) where: r sub e = E[(U+I)/Net Worth], the operating return on net worth. r sub f = the nominal risk free rate.3 U = underwriting income. I = investments/income.

If the effect of changes in interest rate is Fisherian then (r sub m - r sub f and B are constant, and delta E(re)/delta r sub f = 1.

Models of profit determination and models of the cycle have usually ignored the multi-product nature of the insurance firm. Biger and Kahane (1978) showed that under CAPM assumptions the financial markets only impose aggregate requirements on markets. Thus, Equation 1 need apply only for the product mix of each particular insurer. Venezian (1983) has shown that under these conditions competitive insurance markets impose a constraint that forces Equation 1 to be true for each individual line. Venezian (1983) further showed that if insurers are concerned about the variance of returns and make decisions on the return of their portfolio as a whole there can be some arbitrariness in allocating expenses and profits to products. Hence analyses of individual lines as independent entities may fail to capture important elements.

Previous studies of the behavior of profits for individual lines have assumed that the error terms in the various lines are uncorrelated. It is quite conceivable that common factors may affect profits for several lines. As an example, errors in forecasting claim costs could be correlated across lines. Thus the error terms for different lines may not be independent. If errors are interdependent the assumption of independence will reduce the power of any statistical test performed using the data.

Estimation Method

To take into account the possibility of interrelationships between the individual lines of insurance Zellner's (1962) seemingly unrelated regression (SUR) technique, a joint generalized least squares procedure, is used in this study. Under suitable assumptions, joint generalized least squares will be efficient when the errors of the "seemingly unrelated" series exhibit correlations such as those that could be expected in the current context. In the SUR methodology the estimates of the variance-covariance matrix among models are used to estimate the parameters of the joint model. The resulting estimates are equivalent to Aitken's generalized least squares estimates, and have the properties of maximum likelihood estimates.(4)

(1)While Doherty and Kang (1988) recognize the importance of investment income their main thrust is to analyze the cyclicality of underwriting margins.

(2)Float (funds generating coefficient) is the average amount of funds that the insurer has to invest as the result of underwriting a dollars worth of premium.

(3)Equation 1 differs from what has become known as the underwriting CAPM in that it incorporates nominal rather than real returns. Kraus and Ross (1982) find that if returns are stated in nominal dollars the only impact of inflation is to increase the required competitive premia.

(4)See Zellner (1963) and Kmenta and Gilbert (1968) for further discussion of the SUR technique.

Data and Model Formation

The data used for this study were taken from Best's Aggregates and Averages, the New York Statistical Tables and data provided by the New York State Insurance Department. Data are examined for 1960 through 1985. These data allow for examination of the most recent cycle of interest. The lines examined in this study were: fire, allied lines, commercial multi-peril, homeowners multi-peril, ocean marine, miscellaneous liability, inland marine, workers' compensation, automobile liability and automobile physical damage. These lines constitute the substantial majority of all premium writings in the United States over the period examined.

The model used for estimation includes first and second order autoregressive operating margin terms and an unanticipated interest rate term. Operating profits were composed of the return from both underwriting and investment income. The underwriting rate of return was equal to (1-combined ratio). The estimate used for investment income was calculated as the product of the funds generating coefficient and the realized yield on short term treasury securities. This provides an estimate of the return on investment that is independent of the amount of risk that individual companies are willing to undertake. Estimates of the funds generating coefficients were obtained by dividing the reserves by line by the premium volume in the line. The average funds generating coefficient ranged from .618 for automobile physical damage, the shortest tail line, to 2.01 for the long tailed miscellaneous liability line.

The average three month treasury bill yield was obtained from the Federal Reserve Bulletin. The risk free proxy is adopted to separate the investment opportunities associated with underwriting in the various lines from the different investment strategies and risk preferences across insurers. This approach was originally developed by Stone (1975) and has been adopted into most models of insurance pricing. The risk free interest rate was divided into anticipated and unanticipated elements by using a first order autoregressive procedure. The risk free rate predicted from the fitted equation was considered to be anticipated. The balance was defined as unanticipated.

The independent variables in this analysis are of two types, lagged values of the dependent variable and an unanticipated interest rate term. The unanticipated interest rate term was set equal to the actual rate minus anticipated rate defined above. In order to allow for secular trends, linear trend was removed from the operating margin, and the detrended operating margin was then used as the dependent variable in the first stage OLS regression in which the independent variables are lagged operating margins and the unanticipated interest rate. Use of such detrended dependent variables may produce an appearance of cycles of periods comparable to the span of the data, but otherwise provides results that are more efficient than those produced by using differencing.(5)

The form of the model used for estimation was: M sub oit = b sub oit b sub 1i M sub oit -1 + b sub 2i M sub oit-2 + b sub 3i UI sub t + e sub i M sub ont = b sub on + b sub 1n M sub ont-1 + b sub 2n M sub ont-2 + b sub 3n UI sub t + e sub n where M sub oit = operating margin in line i at time t (detrended). UI sub t = the unanticipated interest rate at time t.

The null hypothesis that the parameters were the same in each line was tested using Zellner's (1963) procedure(6).


The correlation matrix for the error terms revealed a high level of interrelationship between the individual lines. Twelve of the 45 pairs of lines had correlation coefficients above .50, and seven of those had correlation coefficients above .60. The results of using the SUR procedure are listed in Table 1.

The results indicate that the model generally explains a large proportion of the variation in the operating margin. The weighted adjusted R(2) for the system is .737. The individual line R squareds are all above .55 with the exception of allied lines. All lines, except allied lines, have significant first or second order autoregressive terms at the .01 level and in most cases both of the coefficients are significant at the .10 level. The standard errors associated with the SUR estimates were generally lower than those from OLS estimation. This is consistent with a gain in efficiency from the use of contemporaneous correlation among lines in the estimation process.

Five lines: homeowners multi-peril, ocean marine, workers' compensation, miscellaneous liability, and automobile liability had positive interest rate variables that were significantly different from both zero and one at the .01 level. Three lines: commercial multi-peril, fire, and allied Lines had positive and significant interest rate terms at .05. The remaining lines had positive interest rate terms that were not significant at .10.

(5)This methodology was used by Venezian (1985).

(6)OLS estimates were also examined for autocorrelation and the maintained hypothesis of no autocorrelation was not rejected.

The interest rate coefficients were generally higher in lines that had higher funds generating coefficients. The Pearson correlation coefficient between the interest rate coefficient and the funds generating coefficient was .8146 with a .0041 probability that the relationship is not significant different from zero.

The analysis performed for testing equality of the parameters across lines produced an F statistic of 2.26 with 27 and 210 degrees of freedom indicating rejection of the equality hypothesis at the .01 level of significance. This indicates that the industry wide approach suffers from aggregation bias and that estimation of models of that form will be insufficient.


The results suggest that treating profit cycles as an industrywide phenomenon results in aggregation bias. The parameters used to describe the autoregressive function in individual line models are significantly different from each other, supporting the hypothesis that lines have individual cycles that differ from each other in period and phase. Analysis that uses industrywide cycles, can be viewed as measuring an average result; such a technique can be misleading since the behavior of larger lines such as automobile liability tend to dominate the results, thus misrepresenting the behavior of lines with smaller premium volumes.

The results also suggest that treating individual lines of insurance as contemporaneously correlated results in more efficient estimation than treating the lines independently. Error terms in some lines of insurance appear to be substantially correlated with those in other lines and using this information in the formulation of the model reduced the standard error of the estimates in most cases.

Finally, the effect of unanticipated interest rates on the profitability of insurers tends to be strong across lines. The relationship between unanticipated interest rates and profitability was significantly different from zero at the .05 level in eight of the ten lines examined. The effect of the interest rate term appears to be related to the length of the loss tail of the lines. Lines with more investable funds are more sensitive to unanticipated changes in interest rates.

Table : Joint Generalized Least Squares Estimates
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Author:Fields, Joseph A.; Venezian, Emilio C.
Publication:Journal of Risk and Insurance
Date:Jun 1, 1989
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