The property/liability insurance cycle: A comparison of alternative models.
Seungmook Choi (*)
Don Hardigree (*) Paul D. Thistle (+) The objective of this paper is to compare alternative models of insurance pricing as theories of the propertyliability underwriting cycle. The existing literature has focused on comparing two models, the financial pricing and capacity constraint models. However, these are not the only relevant models. We show that six alternative models imply the same general form of the pricing equation. We apply the model to data on stock propertyliability insurers for the period 19351997. We find that the actuarial model and the capacity constraint hypothesis are the only theoretical models that are consistent with the data. 1. Introduction Property/liability insurance markets alternate between hard and soft markets in a phenomenon known as the underwriting cycle. In soft markets, underwriting standards are relaxed, prices and profits are low, and the quantity of insurance increases. In hard markets, underwriting standards become restrictive, and prices and profits increase. There are many policy cancellations or nonrenewals, and policy terms (deductibles and policy limits) are tightened as the quantity of insurance coverage generally decreases. (1) The increases in insurance premiums and decreases in the availability of insurance can be sufficiently sudden and severe that hard markets are sometimes referred to as "liability crises." Between 1984 and 1986, industry premium revenue nearly tripled, and availability problems were widely discussed in the popular press, especially in general and municipal liability and in medical malpractice. Following a hard market, prices and profits remain high, then gradually erode as the market softens. The mos t recent hard markets were during 19751977 and 19841987. The consensus estimate is that the cycle has a period of about six to eight years. On this view, at least one hard market should have occurred in the early to mid1990s and possibly another in the late 1990s. However, market conditions have remained soft since the late 1980s. Why the underwriting cycle occurs and whether the cycle continues to characterize insurance markets are questions that do not have complete answers. Both soft and hard markets can create problems. Soft markets may contribute to insolvencies if insurers price and underwrite too aggressively, while high prices and restrictive underwriting in hard markets may disrupt the flow of goods and services. Since the insurance industry is subject to regulation at the state level, understanding the behavior of property/liability insurance markets is important for the development of appropriate public policies. Over the last decade, a substantial body of insurance literature has developed attempting to explain the recurrence of hard markets in property/liability insurance. Most recent studies view the recurrent hard markets as the result of an equilibrium or marketclearing process. The existing literature has focused on comparing two models of insurance pricing, the financial pricing/rational expectations model and the capacity constraint model, as a possible explanation of underwriting cycles. However, since any model of insurance pricing has implications for the underwriting cycle, these are not the only relevant models. Some models (e.g., the actuarial model) are not specifically intended as models of the underwriting cycle, while others (e.g., the capacity constraint model) are. But any model of insurance prices implies that the price depends only on certain variables. Consequently, each model implies that cycles in premiums and profits depend only on those same variables. The objective of this paper is to comp are the alternative insurance pricing models as theories of the propertyliability underwriting cycle. Financial models of insurance pricing based on discounted cash flows and the capital asset pricing model imply that prices equal policy expenses plus the expected present value of claims (e.g., Myers and Cohn 1987). Cummins and Outreville (1987) assume that insurers have rational expectations regarding claims costs and argue that contracting and informational features of the market, regulatory lags, and financial reporting procedures create an apparent insurance cycle (see also Doherty and Kang 1988). These models implicitly assume that capital markets adjust quickly. The capacity constraint model (Winter 1988, 199 la, 1994; Doherty and Posey 1993; Gron 1994a, b, 1995; Doherty and Garven 1995) is based on the assumption of capital market imperfections. The slow adjustment of capital implies that both hard and soft markets will persist. (2) These authors provide empirical evidence that is consistent with the capacity constraint hypothesis. (3) The financial quality hypothesis (Harrington and Danzon 1994; Cagle and Harrington 1995; Cummins and Danzon 1997) extends the capacity constraint model by allowing insurers' default risk to be endogenous. The contingent claims analysis or option pricing (OP) model is a different financial approach to insurance pricing in which the insurance policy is viewed as analogous to risky debt; this model has different implications for the underwriting cycle. (4) There are also nonfinancial models of insurance pricing. In practice, actuarial models are widely used in insurance pricing. In the basic actuarial model, the price of a policy is equal to the expected cost of the policy (expected claims costs plus policy expenses) plus a "risk loading" or "buffer fund" chosen to hold the probability of default to a predetermined level. The wellknown Sandmo (1971) and Leland (1972) model of the competitive firm under uncertainty can also be applied to the property/liability insurance market. The SandmoLeland model is based on the assumption that the firm acts as if it is a riskaverse expected utility maximizer and implies that the price of a policy equals the expected cost of the policy plus a risk premium. The objective of the paper is to compare these six alternative insurance pricing models and determine which of the models is supported by the data. We show that these models imply the same general form for the pricing equation. However, they have different implications for the determinants of prices in the short run and long run. Both short and longrun analyses are necessary to distinguish between the models. Special care is required in the development of the empirical model since some of the variables in question are stationary while others are nonstationary. We examine the possible cointegrating relationships among insurance prices, interest rates, and surplus. Taking the longrun relationships into account, we then examine the shortrun dynamics of insurance prices. Several of the models imply that, in the short run, insurance prices depend on the interest rate and a risk premium. The risk premium in turn depends on surplus and the conditional variance of the loss distribution. Various researchers have p rovided empirical evidence in support of the financial pricing and capacity constraint models as alternative models of the underwriting cycle. This paper considers a wider range of models and provides the first direct empirical comparison of these models. This paper is organized as follows. In the next section, we discuss the alternative models. In section 3, we develop the empirical model and present the empirical results. Our conclusions are offered in section 4. 2. Alternative Models of Insurance Pricing In this section, we provide brief descriptions of the alternative models of insurance pricing. We focus on the main implications of the models for longrun equilibrium relationships and for the shortrun dynamics of prices. We assume that there is a representative firm that operates in a single period. At the beginning of the period, the firm issues q insurance policies, which are sold at price P. The firm pays policy expenses (e.g., commissions, marketing expenses) of c per policy. The firm may invest beginningofperiod resources at the interest rate r. During the period, policyholders realize losses and file claims with the insurance company. Letting x denote the average claim per policy, policyholders' total claims are xq and are paid at the end of the period. The average claim per policy has mean [micro] and variance [sigma] (2.5) We assume throughout that insurers have rational expectations with respect to losses. Letting p = P  c denote price net of expenses, the present value of underwriting profit is [pi](q) = pq  xq/(l + r). (1) The underwriting profit is random since the average claim per policy is random. The financial pricing model is based on the assumptions that the firm is risk neutral and has rational expectations with respect to losses. It also assumes that markets are perfect; in particular, financial capital adjusts quickly and at zero cost. Then the premium net of expenses is equal to the expected present value of losses, p = [micro]/(1 + r). (2) The price is a decreasing function of the interest rate and does not depend on any other variables. The model is illustrated in Figure 1, where the supply of insurance is perfectly elastic in both the short and the long run. The capacity constraint hypothesis (CCH) is a model of shortrun price determination. As a consequence of limited liability, insurers must hold equity to maintain the ability to pay claims, to keep the probability of bankruptcy low, and to meet regulatory requirements. The CCH assumes that there are costs to adjusting the firm's equity and that external equity is more costly than internal equity. (6) Firms hold excess capacity to reduce the probability of becoming capacity constrained in the future, so that soft markets persist. In a hard market, the adjustment costs of raising new equity, combined with the expectation that the bard market will be temporary, implies that firms accumulate surplus internally (Winter 199 la, pp. 1267). The capacity constraint model is also illustrated in Figure 1, where the shortrun supply curve [S.sub.0] assumes a "normal" level of surplus. Given this level of surplus, the capacity constraint becomes binding at [q.sub.0]. As the capacity constraint is approached, the price of the policy must rise in order to maintain an acceptable probability of bankruptcy. At a sufficiently high price, premium revenue is always sufficient to cover claims. An increase in the degree of uncertainty increases the price of insurance in the short run, as higher prices are required to maintain the ability to pay claims (Winter 1988, p. 484). A negative shock to surplus shifts the shortrun supply curve to [S.sub.1], and the price increases to [P.sub.1]. Firms earn positive profits at [P.sub.1], so the capacity constraint can be binding temporarily. As surplus accumulates over time, the market returns to longrun equilibrium. The capacity constraint model implies, in the short run, that prices are decreasing in the interest rate and surplus and increasing in the variance. The longrun supply curve is [S.sub.LR]; that is, the financial pricing model holds in the long run. The CCH implies that the price of insurance can be written as p = [micro]/(1 + r) + R, (3) where R(S, [[sigma].sup.2]) is the shortrun deviation from the longrun equilibrium. Thus, there is a level of surplus, [S.sup.*], above which R = 0 and the price of a policy is just equal to the expected present value of claims. In terms of Figure 1, [S.sup.*] is the level of surplus that leads to the capacity constraint at [q.sub.0]. If surplus falls below this level, then the capacity constraint is binding, and the price rises; that is, if S < [S.sup.*], then R > 0. In the long run, R = 0. The financial quality hypothesis (FQH) assumes that the firm is risk neutral but faces a solvency constraint that premium revenues plus surplus must be sufficient to pay claims: [x.sup.*] = (1 + r)(p + S/q) [greater than or equal to] x. (4) In the FQH, the degree of insolvency risk is endogenous and large enough to affect prices. The firm's objective is to maximize expected profit conditional on solvency, E{[pi](q) + S + K\x [less than or equal to][x.sup.*]}  (1 + r)S, where K denotes bankruptcy costs (Harrington and Danzon 1994; Cagle and Harrington 1995). (7) The FQH also assumes that the demand for insurance depends on the firm's financial quality or default risk. (8) This implies that shocks to surplus shift both demand and supply in the short run. The effect of the supply shift is greater than the effect of the demand shift, and a negative shock to surplus leads to an increase in price in the short run (Cagle and Harrington 1995, p. 227). The financial quality model also implies that price depends on the riskiness of the loss distribution, but the effect is ambiguous. (9) The financial quality model implies that the firm has an optimal capital structure. Higher levels of surplus imply higher levels of financial quality and a greater willingness to pay on the part of consumers. The firm equates the marginal revenue from being able to charge higher prices to the marginal cost of holding additional surplus. Then firms with higher levels of surplus command higher prices. This implies that prices and surplus are positively correlated in the long run. (10) In the FQH prices are given by Equation 3, but now R has two components. The first component, [R.sub.s] is the shortrun disequilibrium component of prices. The second component, [R.sub.L], is the premium for financial quality and is a longrun component of the price. In practice, insurance rates are based on actuarial principles. The primary concern in actuarial pricing models is with setting prices to achieve a predetermined ruin probability or probability of insolvency. Actuarial pricing models imply a risk premium arising from the need for a buffer fund to achieve an acceptable ruin probability. To see this, suppose that q is large enough that the central limit theorem applies, and let [Z.[alpha]] be the upper [alpha] point of the standard normal distribution. Then the premium is again given by Equation 3, but where R = [Z.[alpha]] [sigma]  S/q is now the buffer fund per policy required to yield a ruin probability of [alpha]. (11) The actuarial model implies that underwriting profits depend positively on the variance of losses and negatively on surplus. The actuarial model is similar to the financial pricing model in that supply is perfectly elastic in the short run, albeit at a higher price. The actuarial model also implies that policyholders bear the cost of bankruptcy risk through the higher price. If the risk premium is positive, that is, if [Z.[alpha]] [sigma] > S/q, then the firm will, at least on average, accumulate surplus. In addition, we would expect that as the result of the normal operation of capital markets, firms would attract capital as long as expected profits are positive. The implication is that in longrun equilibrium, the risk premium will be driven to zero, and the premium will equal the expected present value of losses. The model can be contrasted to the OP or contingent claims analysis approach to insurance pricing. The OP approach is based on the recognition that, since insurers have the option to default, an equity position in the insurer can be characterized as a call option on the assets of the firm. Letting C denote the value of the call option, this implies that C(S + pq, r, [[sigma].sup.2]) = S; this must hold in both the short and the long run. The insurer's option to default implies that insurance policies have the characteristics of risky debt. Accordingly, even in the simplest setting, premiums are equal to policy expense plus the expected present value of claims less an adjustment for default risk. That is, policyholders hold a short position in a put option on the assets of the insurer with an exercise price equal to aggregate losses. The value of this put option is B(r, S, [[sigma].sup.2]). Premiums are once again given by Equation 3, where now R = B/q and where B is the value of the insolvency or bankruptcy put option. (12) In the OP approach, it is policyholders rather than the firm that is compensated for risk bearing. The value of the bankruptcy put is a decreasing in the firm's surplus and increasing in the variance of losses. It follows that price is increasing in surplus and decreasing in the variance of losses; however, the effect of the interest rate is ambiguous. (13) All the foregoing models assume that the firm is risk neutral. An alternative model is the wellknown Sandmo (1971) and Leland (1972) model, which we refer to as the economic model. The firm is assumed to be risk averse with utility function u, where u' > 0, u" < 0. (14) The firm's objective is to maximize the expected utility of terminal wealth, [max.sub.q] E{u[(1 + r)(S + [pi](q))]}. The firstorder condition is E{u'(.)[p  x/(1 + r)]} = 0. (5) We assume that the maximization problem has a unique interior solution. Risk aversion implies that the firm must be compensated for bearing risk. That is, the price of the policy must strictly exceed the expected cost of the policy, including both policy expenses and the expected present value of claims, by an amount sufficient to compensate the firm for the risk of the policy. Then in the short run, price is again given by Equation 3; R is a strictly positive risk premium and is increasing in the degree of riskiness and decreasing in the amount of surplus. It can be shown that the model implies that prices and underwriting profits are decreasing in the interest rate. In the long run, remaining in the market must yield at least as much expected utility as exiting. The participation constraint is E{u[(1 + r)(S + [pi](q*))]} [greater than or equal to] u(S). (6) This constraint holds as an equality in longrun equilibrium and can be interpreted as implicitly defining normal expected profits. If the firm is strictly risk averse, the participation constraint implies that expected profits are strictly positive. Since the firm is risk averse, normal expected profits must include compensation for the risk it bears due to the random nature of policyholders' claims. Thus, assuming that it is binding, the participation constraint implies that prices are again given by Equation 3, where the risk premium R depends on the interest rate, surplus, and the variance. All the models imply the price of insurance can be written in the general form p = [micro]/(1 + r) + R(r, S, [[sigma].sup.2]) (7) in both the short and the long run. We carry Out the empirical analysis using the economic loss ratio (Winter 1994). The economic loss ratio is the ratio of an estimate of discounted losses to premiums net of expenses and is a reciprocal measure of price. For convenience, the empirical implications of these models for the economic loss ratio are summarized in Table 1. The "[+ or ]" indicates that a model implies that the variable is a determinant of prices and profits but that the comparative statics effect is ambiguous. A "0" indicates that the variable is not a determinant of prices and profits. An important question is whether these models have different implications for the underwriting cycle; that is, are the models observationally equivalent? Two models are observationally equivalent if all cells in Table 1 are the same. Examining the shortrun implications, the actuarial model, CCH, and economic model are all observationally equivalent, and the shortrun implications of the FQH are similar. However, the longmn implications of the economic model and the FQH are different from each other and from the actuarial and GCH models. The actuarial model and the CCH are observationally equivalent. No other two models are observationally equivalent. It is important to recognize that both the long and the shortrun implications of the models must be examined in order to distinguish among the models. 3. Empirical Analysis In this section, we develop the empirical model based on the theoretical analysis of property liability insurers. Our objective is to construct an empirical model that is sufficiently general to encompass the alternative models of the insurance cycle. We then apply the model to aggregate data on stock property/liability insurers for the period 19351997. In order to carry out the empirical analysis, we need data on the price of insurance, the interest rate, and measures of underwriting capacity or surplus. We measure the price of insurance using the economic loss ratio (ELR), which is the ratio of an estimate of discounted losses to premiums net of expenses (Winter 1994). In principle, we need data on cash flows of claims paid to estimate discounted losses. (15) However, these data are not available for the full sample period. We follow the procedure in Winter (1994) to estimate this ratio from the available data on undiscounted losses. The ELR is estimated as ELR = D X LR/(1  ER), where ER is the expense ratio, LR is the loss ratio, and D is the discount factor. The expense ratio is the ratio of expenses to premiums written, and the loss ratio is the ratio of calendar year losses to premiums earned. (16) In our notation, these measure cq/Pq and xq/Pq, so that LR/(1  ER) measures xq/pq. To estimate discounted losses, let [[beta].sub.s] be the fraction of cl aims that are paid in the sth year after a policy is written. Assuming that [beta].sub.s] is constant over time, then D = [SIGMA] [beta].sub.s]/[(1 + r).sup.s] is an estimate of the present value of discounted claims as a fraction of premiums. The estimates of [[beta].sub.s] are taken from Winter (1994, p. 414) and assume that all claims are paid in eight years.(17) We use the threemonth Tbill rate (TBR), the three to fiveyear government note rate (MOB), and the longterm government bond rate (LOB) as discount rates to construct the variables ELR1, ELR2, and ELR3, respectively. Three measures of surplus are used. The first is the ratio of policyholders' surplus to premiums written (BCAP), the second is the ratio of surplus to assets (BSA), and the third is a measure of relative capacity (RC). Relative capacity is measured as the ratio of surplus to the average surplus over the previous five years, [RC.sub.t] = [S.sub.t]/[[SIGMA].sup.5] [S.sub.tj]. Winter (1994) interprets RC as the cyclical component of surplus. For all three variables, surplus is measured as of the beginning of the year. The insurance industry data are taken from Best's Aggregates and Averages. The interest rate data are taken from the St. Louis Federal Reserve Bank's FRED database, the NBER Macrohistory database, and the Federal Reserve Bulletin. Previous empirical work has found that prices follow an AR(2) process. An important question is why premiums are serially correlated. Suppose that the financial pricing/rational expectations model holds. Then prices are [p.sub.t] = E {x,\[I.sub.t]}/(1 + r), where [I.sub.t] is the information set at time t. If losses are i.i.d. from some distribution with mean [mirco], prices are given by [p.sub.t] = [micro]/(1 + r), and the ELR should not be serially correlated. Suppose instead that losses follow some stationary AR process [x.sub.t] = [phi](L)[x.sub.t1]) + [e.sub.t], where L is the lag operator and the [e.sub.t] are i.i.d. (0, [[sigma].sup.2.sub.e]. If the information set contains [x.sub.t1], then [p.sub.t] = [phi](L)[x.sub.t1]/(1+r). That is, the serial correlation in losses is transmitted into prices, and both the numerator and the denominator of the ELR will be serially correlated. (18) In addition, the fact that policies are issued over the course of the year implies that calendar year accounting data w ill contain information on the current and previous year's pricing decisions. This implies additional serial correlation. That is, if losses follow an AR(1) process, [phi](L) = [[phi].sub.1]L, then the reported accounting data will follow an AR(2) process. This argument also holds if losses have a unit root (i.e., [[phi].sub.1] = 1) as in Cummins and Outreville (1987). The conclusion is that serial correlation in prices can arise from the combination of the stochastic process of losses and insurance accounting practices. Serial correlation in premiums is not per se evidence of informational inefficiencies or market imperfections. If the financial pricing model holds, then the ELR should be stationary and serially correlated. (19) Table 2 reports the results of augmented DickeyFuller tests for unit roots. Panel A of Table 2 reports the results of tests assuming no time trend, and Panel B reports the results of tests assuming that there is a linear time trend. For all three price series, ELR1, ELR2, and ELR3, the unit root hypothesis is rejected against the alternative with and without the trend. The result that the prices series are I(0) implies that premiums are cointegrated with the discounted value of losses. All three interest rate series, TBR, MGB, and LGB, have unit roots. Two of the surplus series, BCAP and BSA, also have unit roots. The third, RC, is stationary around a positive linear time trend. Tests on log transformations of the variables yield the same conclusions. The economic model, the FQH, and the OP model all imply that surplus is a longrun determinant of prices; that is, prices should be cointegrated with surplus. A necessary condition for two variables to be cointegrated is that they both must be I(1). The three ELR series are all I(0), and it follows that they cannot be cointegrated with surplus. The economic model, the FQH, and the OP approach are not consistent with the results of the unit root tests. The results here are consistent with those of Choi and Thistle (1999) and Higgins and Thistle (2000), who find that surplus is not a determinant of underwriting profits in the long run. [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1][DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [[epsilon].sub.t], (8) We now turn to the analysis of the shortrun dynamics. The theoretical models imply that the shortrun dynamics are driven by the interest rate, surplus, and conditional variance. We begin with the regression where y, is the ELR in period t. There are two reasons to include the change in the interest rate in the regression. First, the general form of the theoretical model of prices, Equation 7, implies that it should be included. Second, the dependent variable is the ratio of an estimate of discounted losses to premiums and is certainly measured with error; inclusion of the change in the interest rate may capture the measurement error. The argument for not including the change in the interest rate is that the interest rate is used in the construction of the dependent variable, and the regression will simply reflect this. This suggests the empirical results concerning the effects of interest rate changes need to be interpreted cautiously. The equation is initially estimated by OLS. We then eliminate variables whose coefficients are not significantly different from zero and reestimate the model. The insurance industry has changed substantially over the period 19351997, suggesting that the model may not be stable ov er the entire sample period. We carry Out Chow tests for structural shifts in 1950, 1966, and 1981 (the 1/4, 1/2, and 3/4 points in the sample) and include dummy variables if the hypothesis of parameter stability is rejected at the 10% level. We carry out the analysis for each of the dependent variables, ELR1, ELR2, and ELR3, with each of the measures of surplus. The interest rate and the measures BCAP and BSA are I(1) and are entered in difference form, while RC is I(0) and is entered in levels. The results of the analysis are reported in Table 3 for the price variable ELR1; the results for the log transformations are similar. The first three columns report the estimates for the full model, and the second three columns report the results following the reduction of the model. The estimates reported in column 6 imply that ELR1 follows an AR(1) process and is not cyclical. The estimates reported in columns 4 and 5 imply that ELR1 follows an AR(2) process where the period of the cycle is between seven and eight years. (20) For all three models in columns 4 to 6, the change in the interest rate is negative. This implies that an increase in the change in the shortterm interest rate leads to higher prices. This is consistent with the OP approach but not with the other models. It is also consistent with the view that the coefficient reflects the definition of the dependent variable or that it is capturing measurement error. (21) The coefficients for the lagged values of the surplus measures [DELTA]BCAP, [DE LTA]BSA, and RC are all positive, implying that increases in surplus growth or surplus lower prices although with a delay. This is consistent with all the models except the financial pricing and OP models. The results for ERL2 and ELR3 and their log transformations are similar, except that the lagged interest rate change is significant when RC is used as the capacity measure. The change in the interest rate is not significant when [DELTA]BCAP or [DELTA]BSA is used as the capacity measure. The results in Table 3 are the most directly comparable to previous studies. The definition of the ELRs, together with the fact that they are stationary, implies that the cointegrating relationships are taken account of in the estimates in Table 3. These results are consistent with those of Haley (1993, 1995), Grace and Hotchkiss (1995), and Choi and Thistle (1999), who find that measures of underwriting profits are cointegrated with interest rates. The models of the shortrun dynamics in Table 3 are "balanced" in the sense that all variables are I(0). A number of previous studies have not treated the econometric issues created by the possibility of unit roots appropriately. First, many studies ignore the possibility of cointegrating relationships among the variables. For models estimated in difference form, this implies that all effects are constrained to be shortrun effects. Also, if the error correction terms are important, these models suffer from omitted variable bias. Examples include Niehaus and Terry (1993) and Fung et al. (1998). The second problem is the use of unbalanced regressions, in which the regressand and regressors are not all I(0). For example, Gron (1994b) and Winter (1994) estimate regressions in which the dependent variable is apparently I(0) but one or more regressors is I(1). This implies (i) that the population regression coefficients on the I(1) variables are zero and (ii) that the estimated regression coefficients on the remaining I(0) regressors have nonstandard distributions, so that inference based on asymptotic normal approximations is invalid. (22) The OLS estimates in Table 3 restrict the variance of the error term to be constant. All the theoretical models except the financial pricing model imply that in the short run the economic loss ratio should depend on the conditional variance through the disequilibrium component or risk premium R in Equation 7. The risk premium depends on the conditional variance, [h.sub.t] = var{[[epsilon].sub.t]/[I.sub.t]}, and this variance changes over time as the conditioning information evolves. Since past errors are contained in this information set, this implies that the errors are generated by an autoregressive conditionally heteroskedastic (ARCH) process. The errors are generated by an ARCH process if [[epsilon].sub.t] = [[zeta].sub.t]/[h.sub.t], (9) where [[zeta].sub.t]i.i.d(0, 1) and [h.sub.t] = h([I.sub.t1], [theta]). Since the theory implies that this conditional variance is a regressor, the appropriate model is an ARCHinmean (ARCHM) model. The statistical properties of the ARCH process depend on the form of the variance equation h([I.sub.t1], [theta]). We employ the GARCH process, for which the variance function can be written as [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][[epsilon].sub.t1] + [[theta].sub.2][h.sub.t1]. (10) We also employ the EGARCH process, for which the variance function can be written as ln([h.sub.t]) = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]\  [square root of ((2/[pi]))] + [[theta].sub.2][[zeta].sub.t1] + [[theta].sub.3]ln([h.sub.t1]), (11) where [[zeta].sub.t] = [[epsilon].sub.t]/h, is the normalized error, assumed to be N(0, 1), and [square root of (2/[pi])] = E{\[[zeta].sub.t]\}. The expression in Equation 11 is linear in the shocks, with slope [[theta].sub.1] + [[theta].sub.2] if the shock is positive and slope [[theta].sub.2]  [[theta].sub.1] if the shock is negative. For both these ARCH processes, the unconditional distributions of the [[epsilon].sub.t] are nonnormal and leptokurtic (thicktailed) with mean zero and constant variance. The [[epsilon].sub.t], are also serially uncorrelated; however, there is dependence through the higher moments. (23) We reestimate the regression in Equation 8 with the addition of the conditional variance or the conditional standard error as a regressor, assuming the GARCH process in Equation 10 or the EGARCH process in Equation 11 for the conditional variance. We tested for time trends and for shifts in the variance equation in 1950, 1966, and 1981. Table 4 reports the results for ELR1, ELR2, and ELR3 and their log transformations using [DELTA]BCAP as the measure of surplus and assuming the EGARCH specification in which the conditional standard deviation enters the mean equation. First, there is some evidence of structural shifts over the sample period with the shifts in the mean equation occurring in 1966 or 1981 and the shifts in the variance equation occurring in 1950 or 1966. For the dependent variables ELRl and ELR2, there is no cycle. For the remaining variables, the estimated period is from 9 to 12 years, which is somewhat longer than most estimates. The interest rate coefficient is negative and not significant in the equations for LNELR2 and LNELR3 but is negative and significant in the equations for the other dependent variables. The effect of [DELTA]BCAP on the ELR is positive; however, it is the lagged rather than the current value, which is significant. The conditional standard deviation is significant and negative in the all the equations. T hese results imply that an increase in surplus growth leads to lower prices and that an increase in the conditional variance leads to higher prices. Taken together, the results for surplus growth and the conditional variance are consistent with the actuarial model and CCH and are not consistent with the financial pricing model or the OP model. The results using [DELTA]BSA and RC as the measure of surplus are reported in Tables 5 and 6. The results are qualitatively similar. In both cases, there is evidence of shifts in the mean equation in 1981 and of shifts in the variance equation in 1950 or 1966. In Table 5, estimated length of the cycle is over 18 years for ELR2 and almost 14 years for ELR3, while in Table 6 the estimated length of the cycle is 13.5 years for LNELR1. In the remaining cases, the estimated cycle length ranges from seven to nine years. In Table 5, the change in the Tbill rate has a significant negative effect on ELR1 and LNELRl; the changes in the medium and longterm interest rates are negative but not significant. In Table 6, the effect of the change in the interest rate is never significant. In Table 5, the capacity measure [DELTA]BSA has a significant positive effect, except in the equation for ELR2. In Table 6, the capacity measure RC has a significant positive effect in the equations for ELR2 and LNELR1; the effect is posi tive but not significant in the other equations. The conditional variance has a significant negative effect for all but one of the equations in Table 5 and for all but two of the equations in Table 6. In the remaining equations, the effect is negative but not significant. Again, the empirical results imply that an increase in surplus or surplus growth leads to lower prices and that an increase in the conditional variance leads to higher prices. The results reported in Tables 4 to 6 assume the EGARCH specification of the variance equation and that the conditional standard deviation enters the mean equation. Using the GARCH specification of the variance equation and assuming the conditional standard deviation enters the mean equation yields qualitatively similar results. Using either specification of the variance equation and assuming that the conditional variance enters the mean equation yields weaker support for a significant effect for the conditional variance. Our view is that the specification using the conditional standard deviation is more appropriate. Taken overall, we find no evidence that an increase in the conditional variance (or standard deviation) leads to lower prices in the short run, as implied by the OP model. We find no evidence that an increase in surplus leads to higher prices in the short run, another implication of the OP model. The financial pricing model implies that neither the conditional variance nor the surplus should affect prices in the short run. We find substantial empirical evidence that an increase in surplus or surplus growth leads to lower prices in the short run and that an increase in the conditional standard deviation increases prices in the short run. The actuarial model and the capacity constraint hypothesis are the only theoretical models that are consistent with the empirical results in both the long and the short run. 4. Conclusion Property/liability insurance markets are subject to an underwriting cycle, alternating between soft markets, with low prices and profits, and hard markets, with sharply higher prices and profits. The insurance literature has tended to rely on institutional and regulatory features specific to insurance markets and capital market imperfections to explain recurrent hard markets. Consequently, the empirical literature has focused on testing between the financial pricing and capacity constraint models. In this paper, we carry out an empirical comparison of six alternative models of insurance pricing as theories of the underwriting cycle using data on stock property/liability insurance companies for 19351997. We show that all the models imply the same general form for the pricing equation. All the models imply that the price of insurance equals the costs of issuing the policy plus the expected present value of losses plus an additional term that, in general, depends in the interest rate, surplus, and the conditional variance of losses. In the long run, the additional term can be interpreted as either a risk premium or a quality premium, depending on the theoretical model. In the short run, the additional term can be interpreted as a risk premium, a quality premium, or the disequilibrium component of prices, again depending on the theoretical model. The role of the conditional variance in this additional term has not been emphasized in the literature, and its presence has not been tested for. It is important to distinguish theoretically between the long and the shortrun implications of the models. Four of the six models have the same implications for the shortrun behavior of prices. This implies that it is important to distinguish econometrically between longrun cointegrating relationships and shortrun dynamics. Both need to be examined in order to test between the models. We employ the ELR as an inverse measure of insurance prices. The ELR is the ratio of an estimate of the present value of losses to premiums net of expenses; three discount rates are used to compute ELRs. Three measures of surplus, the ratio of surplus to premiums written, the ratio of surplus to assets, and the ratio of surplus to a lagged moving average are used. We find that the ELRs do not have a unit root. This implies (i) that premiums are cointegrated with discounted losses and (ii) that the ELRs cannot be cointegrated with surplus. Since both the economic model and the financial quality hypothesis imply that there should be a longrun relationship between premiums and surplus, these models are not consistent with the data. We find substantial empirical evidence that an increase in surplus or surplus growth leads to lower prices in the short run and that an increase in the conditional standard deviation increases prices in the short run. The financial pricing model implies that premiums should not depen d on surplus or the conditional standard deviation. The contingent claims analysis model implies that price should increase with surplus and decrease with the conditional standard deviation. These models are not consistent with the data. The actuarial model and the capacity constraint hypothesis are the only theoretical models that are consistent with the empirical results in both the long and the short run. These results have implications for public policy toward the insurance industry. First, and perhaps most important, in the long run, premiums are determined by policy expenses and the expected present value of claims. One implication of this result is that hard markets, such as the episodes in the 1970s and 1980s, are shortrun phenomena. Another implication of this result is that legal and regulatory decisions that increase insurers' claim costs will ultimately lead to higher prices for insurance. Second, surplus is an important determinant of prices in the short run, with higher levels of surplus or surplus growth leading to lower prices. Much of insurance regulation is intended so that insurers are adequately capitalized. To the extent that it is successful, such regulation may lead to lower prices, although only in the short run. Finally, we find that the conditional variance is also an important determinant of prices, with higher conditional variances leading to higher prices. An important implication of this result is that policy decisions that increase insurance market volatility lead to higher prices for insurance, at least in the short run. (*.) Department of Finance, University of Nevada, Las Vegas, 4505 Maryland Parkway, Box 456008, Las Vegas, NV 891546008, USA. (+.) Department of Finance, University of Nevada, Las Vegas, 4505 Maryland Parkway, Box 456008, Las Vegas, NV 891546008, USA: Email pthistle@ccmail.nevada.edu; corresponding author. The authors would like to thank Mast Higgins, James Ligon, Don Meyer, Alan Schlottmann, Mary Riddel, and Brad Wimmer for constructive comments on earlier versions of this paper as well as two anonymous referees whose comments have substantially improved the paper. The authors acknowledge financial assistance from First Interstate Bank Institute for Business Leadership. Thistle's research was partially supported by the Faculty Research and Creative Activities Support Fund, Western Michigan University, while he was on the faculty there. Our friend and colleague Don Hardigree passed away in December 1997. Received October 1998; accepted July 2001. (1.) See, for example, Winter (1991a, pp. 11820) for a description of the typical features of a hard market and Harrington (1988) for a discussion of the financial performance of the property/liability insurance industry focusing on the 19751977 and 19841987 periods. Cycles are not unique to the U.S. insurance industry. See Cummins and Outerville (1987), LammTennant and Weiss (1997), and Chen, Wong, and Lee (1999) on international comparisons. (2.) Winter (1991b) argues that solvency regulation based on premium/surplus ratios exacerbates the problem. (3.) See also Niehaus and Terry (1993), Haley (1993, 1995), Grace and Hotchkiss (1995), Fung et al. (1998). Choi and Thistle (1999), and Higgins and Thistle (2000) for additional empirical evidence. (4.) Sommers's (1996) crosssectional results are consistent with the OP model. (5.) The distribution of losses per policy does not depend on the number of policies sold; that is, the firm sells enough policies to eliminate idiosyncratic risk. The randomness in the average claim per policy arises from dependence in losses across policies. As long as losses are less than perfectly positively correlated across policies, the perpolicy variance will decrease with the number of policies to some limit, That is, losses must obey a law of large numbers (see, e.g., Marshall 1974). (6.) These adjustment costs arise from asymmetries in the tax code (due to the taxation of dividends) and from asymmetric information between management and the market regarding the future prospects of the firm. See, for example, Winter (1988, pp. 4725) for a more complete discussion. (7.) In Harrington and Danzon and in Cagle and Harrington, [KAPPA] is the value of the firm's intangible capital or franchise value that is lost as a result of bankruptcy. Intangible capital includes, for example, the value of reputation and the quasirent on renewal business. (8.) Harrington and Danzon (1994) focus on possible reasons for underpricing in the soft market phase of the cycle. They allow default risk to be endogenous but do not assume that demand depends on default risk. (9.) Cagle and Harrington (1995) do not derive the effect of a seconddegree stochastically dominating increase in the riskiness of the loss distribution. The sign of the effect depends on the third derivative of the objective function, which is indeterminate. Cummins and Danzon (1997) allow for two classes of policyholders and analyze the problem in an option theoretic framework. Their model implies that output depends on equity and the riskiness of the loss distributions but that the comparative statics effects are ambiguous. (10.) The CumminsDanzon. version of the FQH was originally developed as a model of crosssectional differences in prices across firms. The usual interpretation of crosssectional analysis is that they capture longrun behavior. As we point out here, the FQH also has implications for both the short and the longrun timeseries behavior of prices. (11.) See, for example, Beard, Pentikainen, and Pesonen (1984, p. 129), who refer to this as the "basic equation." The number of policies to be sold is exogenous; that is, the firm's demand is perfectly inelastic. Initial surplus is assumed to be exogenous, and surplus is accumulated only through retained earnings. The target probability of insolvency is also exogenous. The idea that the firms might choose to issue new equity or might pay dividends is simply absent from the analysis. (12.) See, for example, Cummins (1991, p. 291, eqn. 54). Policyholders receive the indemnity owed by the insurer if the insurer is solvent; otherwise, they receive the insurer's assets, that is, min(A, xq), where A is assets. Observe that min(A, xq) = xq  max(xq  A, 0) and that the second term is the terminal payoff to a put option. The financial pricing model can be viewed as the special case of the OP approach in which there is no default risk, that is, B = 0. Also observe that there is an implicit assumption in the OP approach that capital markets are perfect. (13.) The effect of a change in the interest rate may be determinate for specific option pricing models. For example, if the assumptions of the BlackScholes model apply, an increase in the interest rate decreases the price of insurance. (14.) Firms will exhibit riskaverse behavior if decisions are made by managers whose compensation and wealth depend on the performance of the firm and who may not be able so optimally diversify; compare Greenwald and Stiglitz (1990). In particular, if managers have firmspecific human capital, reputations, or shares in the firm, then the manager's wealth will be a concave function of the firm's performance. (15.) In principle, we need data on accident year losses, which measures losses arising out of all accidents in a given year. These data are not available for the full sample period. In practice, we use data on calendar year loss, which measures all transactions during a given year. Calendar year losses are equal to accident year losses plus an adjustment for outstanding claims, both reported and unreported. See Heubner, Black, and Webb (1996, p. 625). (16.) loss ratio is the ratio of incurred losses to premiums earned. Incurred losses is an estimate of claims payments associated with policies in force during the year, plus any adjustments to outstanding claims (both reported and incurred but not reported) on expired policies; estimates of future claims payments are not discounted. The expense ratio is the ratio of expenses incurred to premiums written. Premiums written are calculated on a cash basis; premiums earned are calculated on an accrual basis. Most policies are for one year, so premiums earned will reflect policies sold in the current and previous years. (17.) Liability insurance has grown in importance relative to property insurance over the sample period, so that the average lag between payment of the premium and claim settlement has increased. Since the [[beta].sub.s] are calculated from data relatively late in the sample period, this suggests that the present value of losses may be underestimated early in the sample period and that the degree of underestimation may change systematically over time. We include a time trend in our empirical analysis to adjust for this potential measurement error. (18.) This is essentially the argument given in Cummins and Outreville (1987), who assume that shocks to losses have a permanent component; that is, losses have a unit root. This is not necessary for prices to be serially correlated. (19.) Inclusion of lagged terms in the regression may also capture any unexplained cyclical influences. (20.) An AR(2) process is cyclical if [[phi].sup.2.sub.1] + 4[[phi].sub.2] < 0, in which case the period of the cycle is 2[pi]/arccos(\[[phi].sub.1]\/2[square root of ([[phi].sub.2])]). (21.) The measurement error arises from insurance accounting conventions that determine how the data are reported. The numerator of the ELR is D X LR, where D is the discount factor and LR is the loss ratio. The loss ratio is the ratio of losses incurred to premiums earned, where premiums earned are calculated on an accrual basis. An increase in interest rates decreases the present value of losses. Even if prices for new policies adjust, premiums earned, which are a combination of current and previous years' prices, will not fully reflect this, and the numerator of the ELR will decrease. Further, this effect is stronger the more rapidly interest rates change. (22.) See Bancrjcc et al. (1993, pp. 1648) for a discussion of this point. (23.) 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[Figure 1 omitted] Table 1 Summary of Alternative Models' Implications for Economic Loss Ratio Variable Interest Rate Surplus Model SR LR SR Actuarial + + + Capacity Constraint (CCH) + + + Economic + + + Financial + + 0 Financial Quality (FQH) + + + Option Pricing (OP) [+ or ] [+ or ]  Variable Surplus Variance Model LR SR LR Actuarial 0  0 Capacity Constraint (CCH) 0  0 Economic +   Financial 0 0 0 Financial Quality (FQH)  [+ or ] [+ or ] Option Pricing (OP)  + + Table 2 Augmented DickyFuller Tests of the Unit Root Hypothesis, 19351997 A. No Time Trend (a) Variable Level Difference Log Log Difference ELR1 3.36 6.23 3.34 6.31 ELR2 3.21 6.51 3.20 6.65 ELR3 3.24 6.69 3.54 6.83 TBR 1.57 5.71 1.53 5.75 MGB 1.27 3.90 1.13 5.12 LGB 1.02 3.43 0.90 3.99 BCAP 1.65 5.48 1.61 5.36 BSA 1.56 4.99 1.61 5.36 RCAP 3.19 5.64 3.17 5.84 B. Time Trend (b) Variable Level Difference Log Log Difference ELR1 4.23 6.17 4.17 6.25 ELR2 4.13 6.47 4.08 6.61 ELR3 4.37 6.64 4.31 6.78 TBR 2.21 5.69 2.08 5.74 MGB 1.34 3.97 0.97 6.83 LGB 1.54 3.41 1.21 4.00 BCAP 1.35 5.57 1.24 5.46 BSA 1.41 5.04 1.35 4.97 RCAP 3.70 5.59 3.63 5.78 (a)The critical values are 2.59, 2.91, and 3.54 for tests at the 10%, 5%, and 1% levels, respectively. (b)The critical values are 3.17, 3.48, and 4.11 for tests at the 10%, 5%, and 1% levels, respectively. Table 3 Estimates of Regression Models for the Economic Loss Ratio, 19351997 OLS Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.t2][y.sub.t2] + [[beta].sub.1][DELTA] [r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1] [DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [[epsilon].sub.t], Variable 1 C 35.362 (4.32) D81  TIME 0.062 (1.79) ELR1(1) 0.708 (5.27) ELR1(2) 0.184 (1.52) [DELTA]TBR 1.252 (3.39) [DELTA]TBR(1) 0.058 (0.14) [DELTA]BCAP 5.424 (1.18) [DELTA]BCAP(1) 9.20 (2.15) [DELTA]BSA [DELTA]BSA(1) RC RC(1) Adjusted [R.sup.2] 0.641 Period 10.470 OLS Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.t2][y.sub.t2] + [[beta].sub.1][DELTA] [r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1] [DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [[epsilon].sub.t], Variable 2 C 25.215 (2.63) D81  TIME 0.229 (2.92) ELR1(1) 0.656 (4.62) ELR1(2) 0.263 (2.19) [DELTA]TBR 1.258 (3.312) [DELTA]TBR(1) 0.030 (0.66) [DELTA]BCAP  [DELTA]BCAP(1)  [DELTA]BSA 23.494 (1.11) [DELTA]BSA(1) 30.366 (2.13) RC RC(1) Adjusted [R.sup.2] 0.635 Period 7.166 OLS Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.t2][y.sub.t2] + [[beta].sub.1][DELTA] [r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1 ] + [[gamma].sub.1] [DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t 1] + [[epsilon].sub.t], Variable 3 C 38.854 (3.76) D81  TIME 0.095 (2.52) ELR1(1) 0.630 (4.81) ELR1(2) 0.157 (1.34) [DELTA]TBR 1.498 (3.97) [DELTA]TBR(1) 0.137 (0.36) [DELTA]BCAP [DELTA]BCAP(1) [DELTA]BSA [DELTA]BSA(1) RC 59.666 (3.28) RC(1) 54.900 (2.84) Adjusted [R.sup.2] 0.675 Period 9.640 OLS Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.t2][y.sub.t2] + [[beta].sub.1][DELTA] [r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1] [DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [[epsilon].sub.t], Variable 4 C 47.359 (5.58) D81 4.873 (3.43) TIME  ELR1(1) 0.662 (5.04) ELR1(2) 0.258 (2.22) [DELTA]TBR 0.725 (1.98) [DELTA]TBR(1) [DELTA]BCAP [DELTA]BCAP(1) [DELTA]BSA [DELTA]BSA(1) RC RC(1) Adjusted [R.sup.2] 0.650 Period 7.297 OLS Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.t2][y.sub.t2] + [[beta].sub.1][DELTA] [r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1] [DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [[epsilon].sub.t], Variable 5 C 45.349 (5.37) D81 4.324 (3.00) TIME  ELR1(1) 0.666 (5.51) ELR1(2) 0.235 (2.02) [DELTA]TBR 0.875 (2.35) [DELTA]TBR(1) [DELTA]BCAP [DELTA]BCAP(1) [DELTA]BSA [DELTA]BSA(1) 29.715 RC (1.63) RC(1) Adjusted [R.sup.2] 0.660 Period 7.723 OLS Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.t2][y.sub.t2] + [[beta].sub.1][DELTA] [r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1 ] + [[gamma].sub.1] [DELTA][S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t 1] + [[epsilon].sub.t], Variable 6 C 40.711 (3.79) D81 3.872 (2.41) TIME  ELR1(1) 0.505 (4.92) ELR1(2)  [DELTA]TBR 1.217 (3.02) [DELTA]TBR(1)  [DELTA]BCAP [DELTA]BCAP(1) [DELTA]BSA [DELTA]BSA(1) RC 53.757 RC(1) (3.25) 48.701 Adjusted [R.sup.2] (2.47) Period 0.681 [y.sub.t] is the economic loss ratio ELR1, t is a linear time trend, [r.sub.t] is the Tbill rate (TBR), and [S.sub.t] is the ratio of surplus to premiums (BCAP), the ratio of surplus to assets (BSA), or the ratio of surplus to a lagged moving average (RC). D81 is a dummy variable taking the value 1 after 1981. Absolute value of coefficient to estimated standard error in parentheses. Table 4 Estimates of ARCHM Regression Models for the Economic Loss Ratio, 19351997 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta.sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1][DELTA] [S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [delta][square root of ([h.sub.t]) +[[epsilon].sub.t],[[zeta].sub.t 1]  ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]  [square root of((2/[pi]))]] +[[theta].sub.2][[zeta].sub.t1] +[[theta].sub.3]ln [h.sub.t1] 1 Variable ELR1 Constant 36.023 (17.3) D66  D81  TIME 0.019 (1.96) [ELR.sub.t1] 0.904 (53.76) [ELR.sub.t2] 0.145 (5.18) [DELTA][r.sub.t] 1.528 (4.25) [DELTA][BACP.sup.t1] 16.895 (4.31) CV 5.976 (16.29) [[theta].sub.0] 1.054 (9.88) [[theta].sub.1] 0.024 (7.77) [[theta].sub.2] 0.183 (4.93a) [[theta].sub.3] 0.491 (33.38) D50  D66  TIME  Adjusted [R.sup.2] 0.701 Period  EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta.sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1][DELTA] [S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [delta][square root of ([h.sub.t]) +[[epsilon].sub.t],[[zeta].sub.t1]  ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]\  [square root of((2/[pi]))]] +[[theta].sub.2][[zeta].sub.t1] +[[theta].sub.3]ln [h.sub.t1] 2 Variable ELR2 Constant 37.502 (5.43) D66  D81  TIME 0.044 (1.25) [ELR.sub.t1] 0.946 (8.65) [ELR.sub.t2] 0.226 (1.28) [DELTA][r.sub.t] 1.504 (2.57) [DELTA][BACP.sup.t1] 17.076 (2.70) CV 5.640 (4.37) [[theta].sub.0] 1.271 (1.72) [[theta].sub.1] 0.072 (2.08) [[theta].sub.2] 0.300 (248.28) [[theta].sub.3] 0.344 (1.04) D50 0.152 (1.52) D66  TIME  Adjusted [R.sup.2] 0.703 Period  EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta.sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1][DELTA] [S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [delta][square root of ([h.sub.t]) +[[epsilon].sub.t],[[zeta].sub.t1]  ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]\  [square root of((2/[pi]))]] +[[theta].sub.2][[zeta].sub.t1] +[[theta].sub.3]ln [h.sub.t1] 3 Variable ELR3 Constant 39.106 (23.71) D66  D81  TIME 0.203 (16.69) [ELR.sub.t1] 0.824 (134.58) [ELR.sub.t2] 0.243 883.19) [DELTA][r.sub.t] 1.931 (2.33) [DELTA][BACP.sup.t1] 14.419 (10.47) CV 6.587 (315.33) [[theta].sub.0] 0.383 (2.72) [[theta].sub.1] 0.052 (1.00) [[theta].sub.2] 0.187 (5.00a) [[theta].sub.3] 0.514 (201.27) D50  D66  TIME  Adjusted [R.sup.2] 0.654 Period 10.810 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta.sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1][DELTA] [S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [delta][square root of ([h.sub.t]) +[[epsilon].sub.t],[[zeta].sub.t1]  ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]\  [square root of((2/[pi]))]] +[[theta].sub.2][[zeta].sub.t1] +[[theta].sub.3]ln [h.sub.t1] 4 Variable LNELR1 Constant 2.204 (73.91) D66 0.023 (2.50) D81  TIME 0.001 (4.86) [ELR.sub.t1] 0.804 (1.58a) [ELR.sub.t2] 0.211 (11.88) [DELTA][r.sub.t] 2.306 (3.78) [DELTA][BACP.sup.t1] 0.158 (3.12) CV 13.741 (10.02) [[theta].sub.0] 4.967 (8.05) [[theta].sub.1] 0.011 (0.32) [[theta].sub.2] 0.128 (82.05) [[theta].sub.3] 0.2660 (2.70) D50  D66  TIME  Adjusted [R.sup.2] 0.775 Period 12.441 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta.sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.1][DELTA] [S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t1] + [delta][square root of ([h.sub.t]) +[[epsilon].sub.t],[[zeta].sub.t1 ]  ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]  [square root of((2/[pi]))]] +[[theta].sub.2][[zeta].sub.t1] +[[theta].sub.3]ln [h.sub.t1] 5 Variable LNELR2 Constant 2.480 (5.49) D66  D81 0.042 (2.82) TIME 0.001 (4.00) [ELR.sub.t1] 0.773 (4.94) [ELR.sub.t2] 0.332 (2.58) [DELTA][r.sub.t]  [DELTA][BACP.sup.t1] 0.099 (1.91) CV 3.15 (4.00) [[theta].sub.0] 2.953 (3.47) [[theta].sub.1] 0.104 (0.874) [[theta].sub.2] 0.337 (9.24) [[theta].sub.3] 0.596 (4.78) D50  D66 0.285 (2.50) TIME  Adjusted [R.sup.2] 0.739 Period 9.367 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta.sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1 ] + [[gamma].sub.1][DELTA] [S.sub.t] + [[gamma].sub.2][DELTA][S.sub.t 1] + [delta][square root of ([h.sub.t]) +[[epsilon].sub.t],[[zeta].sub .t1]  ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t 1]\  [square root of((2/[pi]))]] +[[theta].sub.2][[zeta].sub.t1 ] +[[theta].sub.3]ln [h.sub.t1] 6 Variable LNELR3 Constant 2.067 (6.41) D66  D81  TIME 0.003 (2.42) [ELR.sub.t1] 0.882 (13.80) [ELR.sub.t2] 0.298 (2.95) [DELTA][r.sub.t]  [DELTA][BACP.sup.t1] 0.158 (3.37) CV 12.190 (3.17) [[theta].sub.0] 4.421 (8.11) [[theta].sub.1] 0.056 (1.34) [[theta].sub.2] 0.058 (2.30) [[theta].sub.3] 0.406 (8.03) D50  D66  TIME 0.006 (1.55) Adjusted [R.sup.2] 0.713 Period 9.969 [y.sub.t] is the economic loss ratio ELR, TIME is a linear time trend, [r.sub.t] is the corresponding interest rate, [S.sub.t] is the ratio of surplus to premiums (BCAP), and [h.sub.t] is the conditinal variance (CV). D50, D66, and D81 are dummy variable staking the value 1 after 1950, 1966, and 1981. Absolute value of coefficient to estimated standard error in parentheses. The "a" indicates times E + 100. Table 5 Estimates of ARCHM Regression Models for the Economic Loss Ratio, 19351997 EGARCHM Modles [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1] [y.sub.t1] + [[phi].sub.2] [y.sub.t2] + [[beta].sub.1] [DELTA][r.sub.t] + [[beta].sub.2] [DELTA][r.sub.t1] + [[gamma].sub.t][DELTA][S.sub.t1] + [delta][square root of ([h.sub.t])] + [[epsilon].sub.t], ln [h.sub.t] = [[theta].sub.0] + [[theta].sub.1][\[[zeta].sub.t1]\  [square root of (2/[pi])]] + [[theta].sub.2][[zeta].sub.t1] + [[theta].sub.3]ln [h.sub.t1] 1 2 3 4 5 Variable ELR1 ELR2 ELR3 LNELR1 LNELR2 Constant 42.755 38.139 36.738 2.436 2.607 (3.53) (275.26) (7.17) (231.64) (256.29) D81 3.928 2.801 1.708 0.032 0.062 (1.58) (6.87) (0.95) (1.83) (3.45) TIME 0.114  0.038 0.002 0.000 (3.73) (1.34) (6.77) (0.66) [ELR.sub.t1] 0.706 0.959 0.940 0.724 0.787 (7.40) (82.27) (34.25) (520.55) (5123.31) [ELR.sub.t2] 0.238 0.258 0.273 0.267 0372 (2.18) (84.78) (3.20) (10.08) (38.19) [DELTA][r.sub.t] 1.159  0.189 1.470 0.001 (2.22) (0.20) (1.97) (0.04) [DELTA][BSA.sub.t1] 42.967 52.859 49.43 0.185 0.104 (2.30) (3.86) (2.88) (1.71) (1.40) CV 2.112 5.214 4.815 4.034 0.849 (1.82) (34.14) (2.92) (1.22) (1.73) [[phi].sub.0] 0.481 0.664 0.669 3.918 12.313 (1.01) (5.84) (1.98) (3.27) (6.48) [[theta].sub.1] 0.055 0.081 0.079 0.067 0.515 (0.36) (4.85) (1.67) (0.60) (1.43) [[theta].sub.2] 0.468 0.158 0.229 0.342 0.115 (1.64) (5.69) (2.41) (147) (0.57) [[theta].sub.3] 0.658 0.646 0.631 0.450 0.991 (3.10) (16.29) (4.19) (2.38) (4.60) D50  1.667 (1.24) D66 0.463  0.329 (1.68) (1.33) Adjusted [R.sup.2] 0.746 0.679 0.698 0.745 0.545 Period 8.24 18.69 13.90 7.91 7.22 6 Variable LNELR3 Constant 2.465 (291.33) D81 0.026 (1.32) TIME 0.002 (4.45) [ELR.sub.t1] 0.818 (2797.62) [ELR.sub.t2] 0.378 (27.55) [DELTA][r.sub.t] 0.021 (0.21) [DELTA][BSA.sub.t1] 0.153 (1.68) CV 3.689 (2.34) [[phi].sub.0] 3.250 (2.48) [[theta].sub.1] 0.085 (1.01) [[theta].sub.2] 0.320 (2.40) [[theta].sub.3] 0.550 (3.02) D50 D66 0.303 (1.84) Adjusted [R.sup.2] 0.729 Period 7.45 [y.sub.t] is the economic loss ratio ELR, t is a linear time trend, [r.sub.t] is the corresponding interest rate, [S.sub.t] is the ratio of surplus to assets (BSA), and [h.sub.t] is the conditional variance (CV). D50, D66, and D81 are dummy variables taking the value 1 after 1950, 1966, and 1981. Absolute value of coefficient to estimated standard error in parentheses; the "a" indicates times E + 100. Table 6 Estimates of ARCHM Regression Models for the Economic Loss Ratio, 19351997 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub.t ] + [[beta].sub.2][DELTA][r.sub.t 1] + [[gamma].sub.t][DELTA][S.sub. t1] + [delta][square rootof ([h.su.t])], + [[epsilon].sub.t], ln [h.sub.t], =[[theta].sub.0] + [[theta].sub.t][\[[zeta].sub. t1]\  [square root of ((2/[pi]))]] + [[theta].sub.2] [[zeta].sub.t1] + [[theta].sub.3],ln [h.sub.t1] 1 Variable ELR1 Constant 34.766 (3.10) D81 4.635 (2.01) TIME 0.077 (1.99) [ELR.sub.t1] 0.739 (91.95) [ELR.sub.t2] 0.230 (2.18) [DELTA][r.sub.t] 0.880 (1.48) [RC.sub.t1] 29.608 (1.55) CV 2.450 (1.44) [[phi].sub.0] 0.366 (0.98) [[theta].sub.1] 0.133 (1.03) [[theta].sub.2] 0.386 (1.32) [[theta].sub.3] 0.717 (4.14) D50  D66 0.334 (1.19) Adjusted [R.sup.2] 0.734 Period 9.08 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.t][DELTA][S.sub.t1 ] + [delta][square rootof ([h.su.t])], + [[epsilon].sub.t], ln [h.sub.t], =[[theta].sub.0] + [[theta].sub.t][\[[zeta].sub.t1 ]\  [square root of ((2/[pi]))]] + [[theta].sub.2] [[zeta].sub.t1] + [[theta].sub.3],ln [h.sub.t1] 2 Variable ELR2 Constant 91.767 (2.39) D81 2.566 (1.61) TIME  [ELR.sub.t1] 0.834 (5.35) [ELR.sub.t2] 0.347 (2.59) [DELTA][r.sub.t] 0.320 (0.58) [RC.sub.t1] 25.972 (1.84) CV 20.464 (1.60) [[phi].sub.0] 3.218 (3.67) [[theta].sub.1] 0.062 (1.71) [[theta].sub.2] 0.011 (0.55) [[theta].sub.3] 0.578 (1.83) D50  D66 0.027 (0.55) Adjusted [R.sup.2] 0.672 Period 8.01 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.t][DELTA][S.sub.t1 ] + [delta][square rootof ([h.su.t])], + [[epsilon].sub.t], ln [h.sub.t], =[[theta].sub.0] + [[theta].sub.t][\[[zeta].sub.t1 ]\  [square root of ((2/[pi]))]] + [[theta].sub.2] [[zeta].sub.t1] + [[theta].sub.3],ln [h.sub.t1] 3 Variable ELR3 Constant 38.676 (947.44) D81 2.319 (0.97) TIME 0.131 (4.32) [ELR.sub.t1] 0.783 (1858.10) [ELR.sub.t2] 0.304 (33.78) [DELTA][r.sub.t] 0.306 (0.20) [RC.sub.t1] 16.392 (1.22) CV 3.327 (3.04) [[phi].sub.0] 0.602 (1.59) [[theta].sub.1] 0.122 (12.80) [[theta].sub.2] 0.362 (2.64) [[theta].sub.3] 0.602 (3.73) D50  D66 0.324 (1.74) Adjusted [R.sup.2] 0.740 Period 8.04 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.t][DELTA][S.sub.t1 ] + [delta][square rootof ([h.su.t])], + [[epsilon].sub.t], ln [h.sub.t], =[[theta].sub.0] + [[theta].sub.t][\[[zeta].sub.t1 ]\  [square root of ((2/[pi]))]] + [[theta].sub.2] [[zeta].sub.t1] + [[theta].sub.3],ln [h.sub.t1] 4 Variable LNELR1 Constant 2.327 (25.200) D81 0.049 (2.32) TIME  [ELR.sub.t1] 0.737 (2290.77) [ELR.sub.t2] 0.170 (4.64) [DELTA][r.sub.t] 0.968 (1.66) [RC.sub.t1] 0.116 (2.41) CV 8.122 (2.06) [[phi].sub.0] 1.768 (4.25) [[theta].sub.1] 0063 (2.46) [[theta].sub.2] 0.105 (1.82) [[theta].sub.3] 0.748 (12.98) D50  D66 0.027 (1.18) Adjusted [R.sup.2] 0.751 Period 13.51 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub.t] + [[beta].sub.2][DELTA][r.sub.t1] + [[gamma].sub.t][DELTA][S.sub.t1 ] + [delta][square rootof ([h.su.t])], + [[epsilon].sub.t], ln [h.sub.t], =[[theta].sub.0] + [[theta].sub.t][\[[zeta].sub.t1 ]\  [square root of ((2/[pi]))]] + [[theta].sub.2] [[zeta].sub.t1] + [[theta].sub.3],ln [h.sub.t1] 5 Variable LNELR2 Constant 2.262 (3.57) D81 0.072 (2.44) TIME 0.001 (1.11) [ELR.sub.t1] 0.791 (7.30) [ELR.sub.t2] 0.310 (2.68) [DELTA][r.sub.t]  [RC.sub.t1] 0.058 (0.93) CV 6.20 (1.75) [[phi].sub.0] 12.613 (19.95) [[theta].sub.1] 0.050 (1.26) [[theta].sub.2] 0.049 (0.81) [[theta].sub.3] 0.976 (12.40) D50 0.609 (1.69) D66  Adjusted [R.sup.2] 0.725 Period 8.04 EGARCHM Models [y.sub.t] = [[alpha].sub.0] + [[alpha].sub.1]t + [[phi].sub.1][y.sub.t1] + [[phi].sub.2][y.sub.t2] + [[beta].sub.1][DELTA][r.sub .t] + [[beta].sub.2][DELTA][r.sub .t1] + [[gamma].sub.t][DELTA][S.su b.t1] + [delta][square rootof ([h.su.t])], + [[epsilon].sub.t], ln [h.sub.t], =[[theta].sub.0] + [[theta].sub.t][\[[zeta].su b.t1]\  [square root of ((2/[pi]))]] + [[theta].sub.2] [[zeta].sub.t1] + [[theta].sub.3],ln [h.sub.t1] 6 Variable LNELR3 Constant 2.443 (205.69) D81 0.023 (1.48) TIME 0.001 (3.97) [ELR.sub.t1] 0.818 (4153.25) [ELR.sub.t2] 0.310 (29.75) [DELTA][r.sub.t]  [RC.sub.t1] 0.041 (0.82) CV 9.655 (3.95) [[phi].sub.0] 3.300 (10.89) [[theta].sub.1] 0.061 (2.22) [[theta].sub.2] 0.128 (3.05) [[theta].sub.3] 0.528 (8.82) D50  D66 0.110 (4.78) Adjusted [R.sup.2] 0.747 Period 8.43 [y.sub.t] is the economic loss ratio ELR, t is a linear time trend, [r.sub.t] is the corresponding interest rate, [S.sub.t] the ratio of surplus to a moving average (RC) [h.sub.t] is the conditional variance (CV). D50, D66, and D81 are dummy variables taking the value 1 after 1950, 1966, and 1981. Absolute value of coefficient to estimated standard error in parentheses; the "a" indicates times E + 100. 

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