# Interest rates, the emergency fund hypothesis and saving through endowment policies: some empirical evidence for the U.K.

Interest Rates, the Emergency Fund Hypothesis and Saving through
Endowment Policies: Some Empirical Evidence for the U.K.

Introduction

Historical trends in most industrial countries indicate that, compared to other forms of financial saving, saving through life insurance has been declining over a long period. Explaining these trends requires examining the factors determining household saving through life insurance. In this connection, various hypotheses have been advanced in the literature, but few have been subject to rigorous empirical testing. (1) This article provides econometric tests of some of these hypotheses for the endowment policies written by British life insurers, using time series data for 1952 through 1985. Apart from these tests, the results also provide estimates of the degree to which saving through life insurance is sensitive to its various determinants.

The model of saving through life insurance used in this study is first outlined. Then empirical results are discussed and hypotheses about the relevance of various factors (e.g., interest rates) that have been suggested in the literature as exerting an important influence on such saving are tested. Households can adjust their saving through life insurance in a number of ways. Typically though, the main effects operate through changes in the allocation of new wealth to life policies and through surrenders. The empirical results based on the model estimated in this study do not, however, allow unambiguous inference about the effects on policy surrenders, which represent the main mechanism via which households adjust their existing holdings of the life insurance asset. Consequently additional evidence is provided on the hypotheses as they pertain to policy surrenders. These findings are, however, specific to one segment of the British life insurance sector: endowment policies. In conclusion, the findings are summarized and their implications discussed.

Saving through Life Insurance: The Model

Various options are available to holders of endowment policies written by British life insurers, in the event they wish to respond to changing capital market conditions by modifying their saving through these policies. They can simply forfeit their policies as they would term policies; they can make the policies paid-up, whereby, even though they cease making premium payments, they retain the benefits these policies offer; they can seek policy loans on the surrender value; or, they can seek a cash surrender.

Two hypotheses in the literature attempt to explain changes in household saving through life insurance, via the mechanisms mentioned above: (i) the interest rate hypothesis, and (ii) the emergency fund hypothesis. The former argues that saving through life insurance is sensitive to rates of return; positively to internal rates of return on endowment policies, and negatively to rates of return on other financial assets. The hypothesis has been examined with respect to policy loans in the United States by a number of researchers [see for example, Hogan (1970), Schott (1971), Pesando (1974), and Cummins (1978)]. The latter hypothesis, originally proposed by Linton (1932), is based on the premise that households regard their savings in life policies as a source of emergency funds which they can draw upon in times of need (e.g., during recession-induced unemployment), either by taking out policy loans or by seeking surrenders. Cummins (1975) tested this hypothesis for the United States for both policy loans and surrenders. His results did appear to support the hypothesis, particularly for surrenders during the 1950s. In the following section, British data on endowment policies are used to examine these hypotheses in terms of the sensitivity of saving through endowment policies to changes in their own rates of return and alternative rates of return, and in a variable constructed to reflect emergency fund considerations.

The Model

There is no unique, integrated theory of saving through life insurance, though several models of the demand for the life insurance product have been developed and used [e.g., Fortune (1973), Headon and Lee (1974), Kindahl (1963), Modigliani (1972), and Neumann (1969)]. There is also controversy surrounding the appropriate definition of saving through life insurance [see Cargill and Troxel (1979)]. The approach in this study is to develop an econometric model on the basis of the Modigliani (1972) stock-adjustment model, which in turn has been used and/or extended by others [e.g., Cummins (1975)]. With regard to the definition of saving, following Cummins (1975), saving through life policies are defined as the net flow of life insurance reserves. The Cummins measure excludes policy loans and examines them separately. The present measure is gross of policy loans, because unlike the general North American experience, policy loans from life insurers in Britain have been very small, and their exclusion in this study is likely to be inconsequential. It might be noted, in passing, that since a homogeneous segment of the life insurance sector is dealt with, the measure of saving is less likely to lead to biases inherent in more aggregated approaches.

The model adopted in this study is a variant of the Modigliani (1972) model and is based on Friedman's (1977) generalization of the basic Modigliani formulation. It can be stated as:

[Delta]A = [Beta] [Delta]W + [[gamma]W.sub.-1] [[Beta] - [b.sub.-1]]

or, in ratio form:

([Delta]A/[W.sub.1]) = [beta] ([Delta]W/[W.sub.-1]) + [gamma][[beta] - [b.sub.-1]

where

A = end-of-period stock of an asset W = end-of-period stock of household wealth b = end-of-period asset-to-wealth holding ratio [beta] = optimum asset-to-wealth holding ratio [gamma] = adjustment parameter (0 < [gamma] < 1) and the minus one subscript denotes a one-period lag.

According to this model, the flow of funds into an asset is made up of two parts: the first representing the allocation of new wealth to an asset according to the optimal holding ratio [beta], and the second reflecting a partial adjustment that involves a reallocation of existing wealth to that asset. The partial adjustment essentially implies that the allocation of existing wealth is more difficult than the allocation of new wealth. In the present context, [Delta]A can be interpreted as the net flow of funds into the life insurance asset; that is, as saving through life insurance. It is measured as the difference between the end-of-period and beginning-of-period stock of reserves. The reserves, as estimated by the Industrial Life Offices Association, are, however, not actuarial reserves, but simply the nominal value of life insurance funds. These funds are, therefore, not the theoretically correct measure of reserves. However, from an empirical point of view, this is not likely to be a serious problem, since the reserves are likely to be measured with error whether or not they are calculated as an actuary would calculate them. Thus the present measure of saving would be subject to measurement error in any case; and as is well known, measurement error in the dependent variable does not affect the quality of even the least squares estimator. (2)

In order to make equation (2) operational, the determinants of the optimal holding ratio [beta] must be specified. On the theoretical grounds, [beta] would, in general, be expected to be directly related to the internal rate of return for life insurance and inversely related to rates of return on competing assets. Whether this is borne out by the data is the central question concerning the validity of the interest rate hypothesis. In this study, instead of using a vector of alternative rates of return, only one such rate of return is considered to capture the effect of the whole spectrum of alternative rates of return.

The use of a single alternative rate of return would appear, at first glance, to be a major limitation. However, this is not the case. The aim is not to construct a consistent household sector portfolio model of asset choice since the objective is not to examine how various assets in a spectrum are related. Rather the focus is on the validity of the hypotheses mentioned above. Indeed, the emergency fund hypothesis need not imply inter-asset substitution at all. In fact, if that hypothesis is valid, it would have qualitatively similar implications for saving through all assets. Secondly, as far as the interest rate hypothesis is concerned, the focus is not on the issue of substitutability between the endowment policies written by the life insurers and each of the other assets. Instead, the study tests whether or not the flow of funds into an asset whose unique feature is saving cum protection is sensitive to market rates of return on other assets in general. For these reasons, the use of a single-alternative rate of return appears to be justified. Additional justification for this approach is provided by econometric considerations. As is well known, high collinearity leads to imprecise and unreliable estimates of parameters in the regression function. The inclusion of several market rates of return which tend to move together, would likely lead to these problems and render tests of the interest rate and emergency fund hypotheses unreliable. Therefore, a single rate of return is used to reflect returns on other assets that compete with the endowment policies.

The internal rate of return raises certain difficulties. First of all, there is no explicit rate of return offered on the life insurance asset considered in this article. There is undoubtedly an implicit yield, but there is no simple way in which that yield can be defined in an aggregate sense. Consequently, the usual approach is either to ignore it, or to use a market-determined interest rate to reflect the aggregate internal rate of return for a sector of the life insurance market. (3) This latter approach is somewhat suspect in that a proxy rate may well reflect the rate of return on competing assets, rather than the internal rate of return for endowment policies. Indeed, this is part and parcel of the collinearity problem alluded to above. Consequently, it would be extremely difficult to meaningfully interpret the interest rate effects when two (or more) such rates are included, one to reflect the rate of return on alternative assets, and one to act as proxy for the internal rate of return. (4) Seen in this light, the exclusion of a proxy rate may not appear to be an altogether inappropriate approach. (5) Though experiments were performed with this approach, the main results are based on a model that explicitly allows for the internal rate of return. This procedure, in fact, is such that the internal rate effects can be estimated without having to actually measure that rate. In this way, not only are collinearity problems circumvented, but also exclusion of a relevant variable with its attendant econometric problems is avoided. (6) With the foregoing discussion in mind, the determinants of the optimal asset holding ratio [beta] ban be specified.

The optimal ratio [beta], is specified as a linear function of the internal real rate of return i and a single real rate of return r reflecting the rate of return on other assets. That is,

[beta] = [[beta].sub.0] + [[beta].sub.1] r + [[beta].sub.2] i

On a priori grounds, the parameters are expected to satisfy the following restrictions: 0 [is less than or equal to] [[beta].sub.0] [is less than or equal to] 1; [[beta].sub.1] [is less than or equal to] 0; [[beta].sub.2] [is greater than or equal to] 0. Thus, if both r and i change by the same amount, then the optimal ratio [beta] will change if ~ [[beta].sub.1] ~ [is not equal to] [[beta]sub.2]. The direction and size of the change in [beta] depends on the size of each parameter, and cannot be predicted a priori.

Cummins argued that inflation could impact adversely on saving through life insurance, because of substitution from life insurance assets to physical assets, and to other saving media like inflation-indexed pension funds, which are better hedged against inflation. However, the effects of inflation on saving are not clear-cut, and there remains considerable uncertainty at both theoretical and empirical levels, about these effects. First of all, in terms of the formulation represented by equation (3), it is clear that if the expected rate of inflation rose, other things being equal, this would lower real estates. However, as stated above, the effect on saving of this equivalent change in real rates of return in ambiguous. The Cummins argument implicitly assumes, in terms of this formulation, that ~ [[beta].sub.] ~ [is greater than] [[beta].sub.2]. That is, the effect of inflation hypothesized by Cummins arises because of the differential impact of insurance's internal and alternative real rates of return on saving. The question that arises is whether inflation has any impact on [beta] that is not grounded in, and independent of, the real rates of return. Thus, if there is a change in expectations about inflation, real rates of return not chancing, would saving through life insurance change? In general, again, this cannot be determined on a priori grounds alone. Thus, if an increase in the expected rate of inflation increases the demand for nominal protection, other things being equal, saving through life insurance could increase of decrease. The answer depends on the distribution of life policies by type, since different policies require different amounts of reserves. In this study, however, only endowment policies are considered. These policies offer protection, but primarily provide an avenue for saving. With real rates constant, the effect of an increase in the expected rate of inflation depends on what happens to the demand for protection through endowment policies. It is possible that households would seek policies geared more protection, in which case saving through endowment policies would decline. However, this remains an empirical question.

On the basis of the foregoing discussion, it follows that:

[beta] = [[beta].sub.0] + [[beta].sub.1.r] + [[beta].sub.2.i] + [[beta].sub.3] [pi]*

where [pi]* is the expected rate of inflation.

Substituting equation (4) into equation (2) and manipulating yields:

([[Delta]A/W.sub.-1]) = [[[gamma]beta].sub.0] + [[beta'.sub.1.r] ([[Delta]W/W.sub.-1]) + [[beta].sub.2.i] ([[Delta]W/W.sub.-1]) + [[beta].sub.3.[pi]* ([[Delta]W/W.sub.1]) + [[[gamma]beta].sub.1.r] + [[[gamma]beta].sub.2.i] + [[gamma][beta].sub.3.][pi]* - [[gamma]b.sub.1] + [mu]X + [[beta].sub.0] ([[Delta]W/W.sub.-1])

where X represents the emergency fund hypothesis whose construction is discussed later.

Assuming that the real rates r and i can be expressed as the corresponding nominal rates r and I less the expected rate of inflation, equation (5) can be written in terms of the nominal rates as follow: (7)

([[Delta]A/W.sub.-1]) = [[[gamma]beta].sub.0] + [[beta].sub.0] ([[Delta]W/W.sub.-1]) + [[beta].sub.1.R] ([[Delta]W/W.sub.-1]) + [[beta].sub.2.I] ([[Delta]W/W.sub.-1]) - [[phi]pi]* ([[Delta]W/W.sub.-1]) + [[[gamma]beta].sub.1.R] + [[[gamma]beta].sub.2.I] - [[[gamma]phi]pi]* - [[gamma]b.sub.-1] + [mu]X

where [phi] = ([[beta].sub.1] + [[beta].sub.2] - [[beta].sub.3]).

In order to make equation (6) operational, three tasks must be completed. First, the nominal internal rate of return I must be eliminated from the equation. Second, how the expected rate of inflation is measured must be indicated. And finally, the emergency fund variable X must be defined.

As stated earlier, the major problem with life insurance's internal rate of return is that it is not an explicit, market-determined rate. However, whatever the implicit yield on endowment policies, market forces will ensure that that yield will be geared to market rates in general, even though evidence for the British life insurers indicates that this implic yield responds only sluggishly to market rates. The relationship between the internal rate I and market rates in general (as represented by R) can be approximated in statistical terms in several ways. Two representations considered in this article are as follows:

I = [[theta].sub.0] + [[theta].sub.1] [(R + [R.sub.1] + [R.sub.2])/3)] I = [[theta].sub.0] + [[theta].sub.1] [R.Sub.-1]

Both equations imply that the internal rate responds only sluggishly to market rates. Clearly, variations of this relationship, all more or less arbitrary, are possible. However, both specifications appeared to perform well, so the estimation is confined to them. If each of (7) and (8) is substituted into equation (6), manipulation yields:

([[DElta]A/W.sub.-1]) = [[gamma]delta] + [delta]([[Delta]W/W.sub.-1) + [[[beta].sub.1 + ([[beta].sub.2].[[theta].sub.1./3])] R ([[Delta]W/W.sub.-1]) + [[[beta].sub.2].[[theta].sub.1./3]] ([R.sub.-1] + [R.Sub.-2])([[Delta]W/W.sub.-1]) - [[phi]pi]* ([[DElta]W/W.sub.-1]) + [gamma][[[beta].sub.1] + ([[beta].sub.].[[theta].sub.1./3])] R + [gamma][[[beta]sub.2].[[theta].sub.1]./3] ([R.sub.-1] + [R.sub.-2]) - [[[gamma]phi]pi]* - [[gamma]b.sub.-1] + [mu] X

where [delta] = [[[beta].sub.0] + [[beta]sub.2].[[theta].sub.0]].

([[Delta]A/W.sub.-]) = [[gamma]delta] + [delta][[Delta]W/W.sub.-1]) + [[beta].sub.1.R] ([[Delta]W/W.sub.-1]) + [[beta].sub.2].[[theta].sub.1.([R.sub.-1])] ([[Delta]W/W.sub.-1]) - [[gamma]pi]* ([[Delta]W/W.sub.1]) + [[[gamma]beta].sub.1.R + [[[gamma]beta]sub.2].[[theta].sub.1].[R.sub.-1] - [[[gamma]phi]pi]* - [[gama]b.sub.-1] + [mu] X

Equations (9) corresponds to equation (7) for R, while equation (10) corresponds to equation (8) for R.

The emergency fund variable X is clearly a proxy variable, since there is no obvious or clear cut definition of what it should be. Two measures were considered, each of which is an unemployment-related construct and is therefore likely to be a reasonably good proxy for capturing the behavior implied by the emergency fund hypothesis. To be sure, other measures which reflect swings in the cyclical position of the economy could serve the same purpose.

The first measure is the annual rate of growth in the level of unemployment. This measure was used successfully by Cummins (1975). If the emergency fund hypothesis is valid, [mu] would be expected to be less than zero. That is, an increase in the rate of growth of unemployment, ceteris paribus, would reduce the net flow of funds into the life insurace asset, as households increased their policy loans and/or sought cash surrenders. With the second measure, X is defined as the level of actual unemployment relative to trend unemployment. Again, this measure would capture any sharp swings in economic activity, and if the emergency fund hypothesis is valid, [mu] would be expected to be less than zero.

Finally, a definition of the expected rate of inflation is considered. Expectations about inflation are assumed to be based on past rates of inflation. A popular model of expectations formation is the adaptive expectations model. (8) As is well known, if economic agents form their expectations of inflation adaptively, this is formally equivalent to defining [pi]* as a geometric weighted average of current and past rates of inflation. In other words:

[[pi].sub.t]* = [Mathematical Expression Omitted]

where [lambda], such that 0 [is less than] [lambda] [is less than] 1, is the weighting parameter. (9) Apart from considering this approach to expectation formation, a simpler definition of [pi]* was also considered. According to the latter definition, the expected rate of inflation was defined as a three-year arithmetic average of current and past rates of inflation. (10)

Estimation and Empirical Results

Several factors were considered in devising the estimation method. (11) To begin with, both equations (9) and (10) which represent the basic model, involve non-linear constraints on the coefficients of the variables, even though those equations are linear in variables. In order to take these restrictions into account (and this is necessary to get unique estimates of the parameters) the equations must be estimated by non-linear methods. (12) Second, it is immediately clear that some parameters in each equation are underidentified. That is, without further restrictions, only [[beta].sub.1] (the alternative rate of return effect) and [mu], the coefficient of the emergency fund variable can be estimated. Both parameters are of central interest in this study. However, [[beta.sub.2], the internal rate of return effect, and less importantly, [[beta.sub.3] and [[theta].sub.1] cannot be obtained. With the restriction: [phi] = [[beta].sub.1] + [[beta]sub.2] (that is, [[beta.sub.3] = 0), however, both [[beta].sub.2] and [[theta].sub.1] become estimatable. This restriction, which is also tested for, implies that inflation does not have an impact on saving through endowment policies that is indepdent of its impact operating through the real rates of return. These equations were estimated with and without this, and other testable restrictions.

The third factor which played an important role in determining the estimation method, was that since most of the right-hand side variables are likely to be endogenous and correlated with the disturbance term in the equations, the least squares estimators are biased, and the bias persists even in large samples. The instrumental variables estimation method was used to deal with this problem. (13) This method involves obtaining a set of variables which are uncorrelated with the disturbance term and using these as instruments for the endogenous variables. An added advantage of this method is that it also deals with the inconsistency of the estimates that results when explanatory variables are measured with error. (14)

Analysis of the Results

It is clear from the foregoing discussion that, given the two alternative definitions of both [pi]* and X, as well as the various parameter restrictions, these are several sets of estimates of eacch of equations (9) and (10). The results were very similar whether the geometric weighted average or the three-year arithmetic average definition of the expected rate of inflation was used. Further, the different definitions of the emergency fund variable X also gave very similar results. If anything, the rate of growth in unemployment definition of X produced equations that appeared to be somewhat more efficiently estimated. In light of this, the results reported below in Table 1 for equations (9) and (10) are based on the arithmetic average definition of [pi]*, and the rate of growth definition of X. (15)

The first two rows in Table 1 are the estimates of equations (9) and (10) with no restrictions. Both equations appear to fit the data well, and as a whole are highly significant at evven the 5 percent level. Further, tests on the Durbin-Watson statistic show that autocorrelation is absent from both equations. Turning to the parameter estimates, the estimate of [[beta]sub.1.], alternative rate of return parameter, not only has the right sign but is also significant at even the 1 percent level. This indicates that saving through endowment policies does respond to changes in market rates of interest. The effect of the internal rate of return [[beta].sub.2], on the other hand, cannot be estimated from these unrestricted equations. This is also true for the parameters [[beta].sub.3] and [[theta].sub.1]. Instead, only [[beta].sub.4] can be estimated, which is a composite of [[beta].sub.2] and [[theta].sub.1], and the results indicate that this estimate is highly insignificant. Estimates of [theta], the unrestricted coefficient of the inflation variable indicate that it is also highly insignificant. Further, there is no support at all for the emergency fund hypothesis, because the estimates of the parameter [mu] are statistically insignificant. The estimates of [gamma], the partial adjustment parameter, are barely significant at the 5 percent level and point to about a 2 percent elimination of the gap between the actual and optimal holding ratio in any period. This implies a slow and sluggish adjustment path to the optimal holding ratio.

In order to estimate the internal rate of return coefficient [[beta].sub.2], some restrictions must apply to the inflation coefficient [theta]. The logical restriction is: [theta] = [[beta].sub.1] _ [[beta].sub.2], or equivalently [[beta].sub.3] = 0. The estimates of equations (9) and (10) obtained with the restrictions, are presented in the third and fourth rows, respectively, of Table 1. In broad terms, the statistical properties of these equations are very similar to those obtained earlier. There is, in fact, only a slight change in the estimated coefficients. Of interest are the estimates of the coefficients of the interest rate variables. The estimate of the alternative rate of return effect [[beta].sub.1] is hardly changed in terms of both magnitude and statistical significance. In addition, estimates of the internal rate of return parameter [[beta].sub.2] are now available. These have the correct (positive) sign and are highly significant. Thus, the evidence suggests that both the alternative and internal rates of return do affect saving through endowment policies, and do so in the directions predicted by theory. However, again there is absolutely no support for the emergency fund hypothesis.

There are certain aspects of these results that merit further exploration. It may be noted that the results from the first pair of unrestricted equations support the restriction that the inflation coefficient [theta] = [[beta]sub.1] + [[beta].sub.2] - [[beta].sub.3] = 0. This in turn suggests one of the following restrictions: [[beta]sub.1] + [[beta]sub.2] = 0 was also tested and overwhelming support for it was found, thus ruling out the third possibility. This, with the implication (from the first pair of unrestricted equations) that [theta] = 0, supports the second set of restrictions. To explicitly test whether [[beta].sub.3] is indeed zero, equations (9) and (10) were estimated with the restriction that [theta] = -[[beta].sub.3], equivalently ([[beta].sub.1] + [[beta].sub.2]) = 0. These estimates are reported in the fifth and sixth rows of Table 1. Apart from some deterioration in fit, the results are very similar to those obtained without this restriction. More importantly, it is clear that the estimates of [[beta].sub.3] are highly insignificant, thus lending clear support to the restriction that [theta] = [[beta].sub.1] + [[beta]sub.2] = [[beta].sub.3] = 0. The estimates of equations (9) and (10) under this restriction are reported in the last two rows of Table 1. These estimates do not alter earlier conclusions about the hypotheses being tested. On the other hand, there appears to be some improvement in the efficiency of the estimates, as one might expect from the imposition of restrictions supported by the data. This improvement is reflected in reduction in the estimated standard errors of a number of estimated parameters. This final pair of equations also exhibits the best fit, and each is highly significant as a whole.

There are some additional aspects of the results that warrant further analysis. First of all, it has been shown that interest rates do matter in terms of their impact on saving through endowment policies. In order to gauge the strength of the interest rate effects, the elasticity of that saving can be obtained with respect to each rate of return. However, given that there are n o data for the internal rate, only the elasticity with respect to the alternative rate of return can be estimated. The expression for this elasticity ([xi]) is: (16)

[xi] = {[[beta].sub.1] [[Delta]W + [[gamma]W.sub.-1]]}R/[Delta]A

Based on the sample average values of the variables appearing in this expression, and on the average estimated values of [[beta].sub.1] and [gamma] in Table 1, the estimate of the elasticity is -1.4. This suggests that saving through endowment policies is interest elastic, that is, highly responsive to changes in alternative rates of return. It needs to be borne in mind, however, that this is a partial elasticity, in that the internal rate of return is not permitted to change when market rates change. Clearly, in practice, one would expect the internal rate to respond (sluggish though that response may be) to market rates of return, so that the total, as opposed to partial elasticity, would likely be smaller.

The second noteworth feature of these findings is the implication [[beta].sub.1] + [[beta].sub.2] = 0. This suggests that endowment policies are a one-for-one substitute with alternative financial assets, in that if both the internal and alternative rates were to change simultaneously by the same amount, there would be no net impact on saving through endowment policies. However, if the internal rate of return does not increase or increases only slowly when market rates increase, a substantial shift of saving from endowment policies to other assets is suggested by the high interest elasticity of such having.

When saving through endowment policies is measured as the net flow of funds into these policies, emergency fund considerations have not effect on such saving. This saving is, however, influenced by changes in the internal and alternative rates of return. Indeed, the elasticity estimate indicates a high responsiveness of saving to the alternative rate of return. Also, endowment policies and other assets appear to be one-for-one substitutes for one another, and inflation does not appear to have any net impact on saving through those policies.

Interest Rates, Emergency Fund Considerations and Policy Surrenders

One of the main insurance mechanisms through with changes in saving occur is a change in surrenders. Further, any change in surrenders will be reflected in a reversal in the net flow of life insurance saving. On the other hand, a change in the latter need not necessarily be associated with a change in surrenders. For instance, the net flow of funds can decrease without an increase in surrenders, if the new funds households wish to put into life insurance should decline for other reasons, give that there are alternatives to a straight surrender. For example, if households decide to forfeit policies or make them paid-up, the net flow would decline. Thus, it would appear these findings about the effects of interest rates and emergency fund considerations on saving through life insurance do not provide unambiguous information about the response of surrenders. In this context, it might be noted that the emergency fund hypothesis, if valid, would operate mainly by affecting surrenders. But this effect may not be captured by relating the net flow of funds into life insurance to the emergency fund variable, as in the previous section, because surreders are but one element in that net flow. It is therefore of some importance to empirically determine whether or not emergency fund considerations, or interest rates, have an independent impact on surreders.

In order to do this, the following model for surrenders is adopted:

S = [[alpha].sub.0] + [[alpha].sub.1.R] + [[alpha]sub.2.I] + [[theta]pi]* + [[alpha].sub.5.V.sub.-1] + [mu] X

where S = surrenders, V = end-of-period stock of life insurance funds, and X is the emergency fund variable measured in either of two ways described in the previous section. As before, R and I are, respectively, the alternative and internal rates of return. The stock of life insurance funds is included to capture the scale effect on surreders, and represents the upper bound on policy surrenders in any period. On a priori grounds one would expect that 0 [is less than or equal to] [[alpha].sub.5] [is less than or equal to] 1. One would also expect that [[alpha].sub.1] [is greater than or equal to] 0, and [[alpha].sub.2] [is less than or equal to ] 0, while [mu] would be positive if the emergency fund hypothesis valid, but zero otherwise. The sign of [theta], the coefficient of the inflation variable cannot be predicted on theoretical grounds alone. If real rates are assumed to be relevant to decision making, then according to the argument of the previous section, the effect of inflation on surrenders, via its impact on real rates, is given by - ([[alpha].sub.1] + [[alpha].sub.2]), and this sum cannot be signed on prior grounds. If inflation also has an impact on surreders that is independent of (in addition to) its impact via real rates, then:

[phi] = - ([[alpha].sub.1] + [[alpha].sub.2] - [[alpha].sub.3.])

where [[alpha]sub.3] is the impact of inflation, real rates of return remaining unchanged. This effect of inflation occurs if inflation has an impact on the demand for protection in general, and to the extent that endowment policies are not as protection-oriented as other policies, [[alpha].sub.3] might be expected to be positive. In any event, it is clear that the unrestricted effect of inflation, as measured by [theta], cannot be signed on a priori grounds. Nor can the parameters [[alpha].sub.2] and [[alpha]sub.3] be identified or estimated.

In order to make equation (11) operational, the internal rate of return variable I must be eliminated. This is done following the procedure outlined in the previous section. Thus, using equation (7) for R, it can be shown that the equation (11) reduces to:

S = [delta] + ([[alpha]sub.1] + [[alpha.sub.4]) R + [[alpha].sub.4] ([R.sub.-1] + [R.sub.-2]) + [[phi]pi]* + [[alpha].sub.5] [V.sub.-1] + [mu] X where [delta] = [[[alpha].sub.0] + [[alpha].sub.2].[[theta].sub.0]], [[alpha].sub.4] = ([[alpha].sub.2].[[theta].sub.1./3]), and [theta] = - ([[alpha].sub.1] + [[alpha].sub.2] - [[alpha].sub.3]). On the other hand, if the approximation is used for I given by equation (8) the estimating equation becomes:

S = [delta] + [[alpha].sub.1.R + [[alpha].sub.4.R.sub.-1] + [[theta]pi]* + [[alpha].sub.5] [V.sub.-1] + [mu] X

A number of points about the estimation of these equations may be noted. First, only equation (12) requires non-linear estimation. Equation (13) has no non-linear restrictions, and can thus be estimated by linear methods. Second, [[alpha].sub.2], [[alpha].sub.3] and [[theta].sub.1] are not identified, and thus cannot be estimated. Finally, each of these equations was estimated using the two definitions of the emergency fund variable, as well as the two formulations of the expected rate of inflation.

In general how expected inflation is measured is not important in terms of the nature of the findings. Different definitions of the emergency fund variable X also gave broadly similar results, with the same general conclusions. One major difference was that while the actual-to-trend unemployment definition of X led to better statistical fits than the annual growth rate of unemployment definition, the former performed rather poorly in terms of autocorrelation even when a correction for it was possible. The latter definition, on the other hand, gave results that did not suffer from this problem. As a consequence, the latter results are likely to be more reliable. These results are reported in the first two rows of Table 2.

It may be n oted that the estimates of equation (12) have been obtained after adjusting for autocorrelation using the Corchrane-Orcutt method. The (adjusted) coefficient of determination declines sharply. However, this statistic is neither comparable to its value in other equations, nor is it very meaningful in the present context. (17) More importantly, it is clear from the Durbin-Watson statistic that neither equation suffers from autocorrelation.

It is clear that only the estimates of the coefficients of the stock variable and the emergency fund variable are statistically significant at the 5 percent level. The estimate of the alternative rate of return effect, as measured by [[alpha].sub.1], is highly insignificant in the first equation, and just barely so in the second. In both equations, this coefficient has the wrong sign, and it is likely that is has not been well estimated in either. The same applies to the estimates of the composite parameter [[alpha].sub.4], as well as the inflation parameter [theta], both of which are highly insignificant. In short, the estimates of equations (12) and (13) provide support for the emergency fund hypothesis, but neither the alternative rate of return nor inflation appear to have any impact on surrenders.

In the versions of the models estimated above, it was not possible to estimate the internal rate of return effect as measured by [[alpha].sub.2], since that parameter is not identified. In order to estimate the internal rate effect, the identifying restriction [[alpha].sub.3] = 0 can be imposed. Under this restriction, [[alpha].sub.1] and [[alpha].sub.2] as well as [[theta].sub.1] can be estimated, and the inflation coefficient [theta] is estimated by the sum: - ([[alpha].sub.1] + [[alpha].sub.2]). However, estimation of the parameters of both equations (12) and (13) requires non-linear methods. The results for these equations, under the afore-mentioned restriction are reported in the last two rows of Table 2.

An examination of the estimates shows that results are quite similar to those obtained earlier. The alternative rate of return effect has the wrong sign, but is highly insignificant in both equations. Again, the stock variable has a significant impact on surrenders, and the emergency fund hypothesis is also supported, as the estimate of [mu] is significant at the 5 percent level. The estimates of the two new parameters, [[alpha].sub.2] and [[theta].sub.1], are both highly insignificant. Thus, surrenders in British endowments do not appear to be affected by interest rates.

It is interesting to note that the above findings on the impacct of interest rates, and on the emergency fund hypothesis are just the opposite of those obtained in the previous section. This suggests that while saving through endowment policies is affected by interest rates, the adjustment takes place through channels other than policy surrenders. This implies that interest rates typically determine how households allocate their new wealth between assets but do not affect existing wealth in the form of endowment policies through surrenders. On the other hand, emergency fund considerations affect saving through endowment policies in just the opposite way.

Conclusions

Empirical findings suggest that saving through endowment policies are responsive to internal returns for British endowments and to alternate rates of return on alternative investments. Furthermore, if both internal and alternative real rates change by equivalent amounts, there is no impact on saving through those policies, thus implying that those policies and other financial assets are, in general, one-for-one substitutes. On the other hand, since intrenal rates respond only sluggish to changes in market rates, one would expect a strong adverse impact on saving through endowment policies if market rates were to increase. (18) Indeed, the declining long-term trend in saving through endowment policies relative to saving in other assets, may well reflect the sluggish response of the internal rate for endowments to generally rising market rates. The interest sensitivity of saving, however, cannot be attributed, even in part, to surrenders. In fact, no significant relationship between surrenders and rates of return was found. Thus, it appears that changes in rates of return affect saving through endowment policies primarily through their impact on the flow of new wealth.

Emergency fund considerations were found to have no influence on saving through endowment policies. On the other hand, surrenders were found to be significantly related to the emergency fund variable. However, since the emergency fund hypothesis is primarily a cyclical phenomenon, these findings imply that emergency fund considerations have a short-run impact on saving through endowment policies, but cannot explain the observed trends relative to other forms of saving. Findings on the rate of return effects appear to be consistent with those trends.

While this study deals only with the endowment policies of British life insurers, findings that saving through those policies are highly sensitive to relative rates of return, and (via surrenders) to emergency fund considerations, may well apply to other segments of the life insurance sector. In that event, the life insurance sector as a whole, could be subject to the type of short and long term effects found for endowment policies. Thus, a period of relatively rising alternative rates of return could sharply shift saving patterns in favor of assets offered by other financial intermediaries. This would imply a shift in the volume and composition of capital flows into bond and equity markets, with consequent wider macroeconomic effects. On the other hand, findings for endowment policies written by British life insurers also point to important distributional effects within the life insurance sector. For example, a shift in saving away from endowment policies (as a result of a change in rates of return) may reflect policyholders' desire to separate protection from saving. This could result in a movement towards protection-only policies, and since any compensating increase in such policies does not generate the same amount of reserves for investment, there could be a decline in the aggregate flow of investible funds.

In conclusion, as is the case in most empirical studies, some caution in warranted in using quantitative results to make inferences. In this context, these results are specific to the endowment policies of British life insurers. Thus, any inferences about the implications of these findings for the life insurance sector as a whole are merely suggestive. On the other hand, they appear to be plausible, and are consistent with observed trends within the British life insurance sector.

Appendix: Variable Definitions and Data Sources

Wealth (W) is defined as the net worth of the personal sector and is calculated as follows: A benchmark estimate of W for 1957 was taken from Revell (1967), p. 163, Table 7.6, and cumulated through 1985 with the help of Britain's personal saving series. The latter data were drawn from various issues of the Blue Book -- National Income and Expenditure, HMSO.

The data for surrenders (S) and for the stock of life insurance reserves (A) were from the Industrial Life Offices' Association (ILOA). Saving through endowment policies were then measured as the difference between end-of-period reserves and beginning-of-period reserves, while b, was measured as: A/W. It might be noted that the estimates of life insurance reserves published by the ILOA are not actuarial reserves but reflect the market value of life insurance funds.

For the rate of return on alternatives assets, the 2.5 percent of War loan (a perpetuity) end-of-year flat yield was chosen. The data for this wate were taken from various issues of the Bank of England Quarterly Bulletin.

The inflation rate ([pi]) was measured as the annual percentage change in the consumer price index, data for which were obtained from issues of International Financial Statistics: Yearbook, the International Monetary Fund.

The instrumental variables chosen (other than those variables that appear in the estimating equations) were the following: (1) exports ([pound sterling] million) (2) unit value index for exports (1975 = 100) (3) foreign exchange reserves (beginning of period -- $ million U.S.) (4) government expenditure ([pound sterling] billion) (5) government consumption ([pound sterling] billion) (6) mid-year population (millions). The data for these were drawn from various issues of International Financial Statistics: Yearbook, The International Monetary Fund.

(1) See, however, Cummins (1975) and Pesando (1974).

(2) See Johnston (1984), pp. 428-30. See also the appendix for data resources, and for definitions of all other variables.

(3) Modigliani (1972) uses the mortgage rate, while Cummins (1975) uses the rate on directly placed bonds in the United States.

(4) The upshot of this is that some market-determined rate of return would better reflect movements in the rates of return on alternative assets, than it would changes in the implicit return on the life insurance asset. This is because alternative rates of return are themselves market rates.

(5) On the other hand, the exclusion of a relevant variable could lead to biased and inconsistent estimates, if the estimation method is not appropriately amended. See Johnson (1984), pp. 259-64. This leads to a rather serious dilemma in that the inclusion of a variable when it is highly collinear with other included variables gives rise to unreliable estimates. It is generally more difficult to deal with the latter problem.

(6) In fact, this estimation method is sufficiently robust to circumvent these econometric problems as well.

(7) Alternatively, R = r + [pi]*, and I = i + [pi]*.

(8) Versions of this formulation have been used extensively is saving/consumption studies. See, for instance, Weber (1975) and Gylfason (1981). If expectations about inflation were adaptive, they could be represented as follow:

[pi]* = [lambda] + (1 - [lambda])[pi]* -1.

where [lambda], such that 0 [is not equal to] 1, is the coefficient of expectation.

(9) This equivalence can easily be shown by using this definition in the equation, and then applying a Koyck transformation. See Johnston (1984), pp. 346-48.

(10) This is undoubtedly a simple formulation, but it has been used with success in savings studies. Some support for using three years comes from the finding that the period adequately defines the time horizon of households [see Friedman (1957), Mohabbat and Simons (1977)].

(11) It has been implicitly assumed that each equation is subject to a disturbance term.

(12) The non-linearity arises because the coefficients are non-linear combinations of each other. Of course, if one were not interested in the individual parameters, the equations could be estimated by linear methods.

(13) See Johnston (1984), pp. 363-66, and 430-32, for this method of estimation. A careful choice of instruments would generally require knowledge of the set of pre-determined variables in a larger model of the British economy. A set of variables has been chosen that is likely to satisfy this requirement. A part of this set is made up of the explanatory variables. The rest of the variables do not explicitly appear in the equations, but are viewed as pre-determined within the context of a larger model, of which this one is a part. These variables are listed in the appendix.

(14) The implications of measurement error are similar to those resulting from the exclusion of a relevant explanatory variables. both lead to the general problem of correlation between the explanatory variables and the disturbance term. See Johnston (1984), pp. 428-35.

(15) All estimation was done using the Time Series Processor (TSP), Version 4.0 computer package.

(16) This elasticity is defined in the following way: [xi] (* log [Delta]A/* log R) = (* [Delta]A/* R) (R/[Delta]A), where the bar denotes the sample mean of the variable concerned.

(17) In this equation, the correction for autocorrelation involves a transformation of the equation, so that the dependent variable is no longer the same. As a consequence, the coefficient of determination is not directly comparable across equations. In addition, no clear meaning can be attached to any measure of goodness of fit, since the transformation is predicated only on econometric, and not theoretical grounds. It is also uncommon for the fit of the transformed model to deteriorate.

(18) The strong impact results from the high interest sensitivity of saving.

A. Dar is Associate Professor and J. C. Dodds is Dean of Commerce at Saint Mary's University, Halifax. An earlier draft of the paper benefited from comments from two anonymous referees and from Dr. S. Amirkhalkhali, however, the authors accept responsibility for any errors.

References

[1.] Cargill, T. F., and T. E. Troxel (1979) "Modeling Life Insurance Savings; Some Methodological Issues," Journal of Risk and Insurance, 46, 391-410.

[2.] Cummins, J. D. (1975) An Econometric Model of the Life Insurance Sector of the U. S. Economy (Lexington, Massachusetts: D.C. Health).

[3.] Fortune, P. (1972) "A Theory of Optimal Life Insurance: Development and Tests," Journal of Finance, 28, 317-26.

[4.] Friedman, B. J. (1977) "Financial Flow Variables and Short-Run Determination of Long-Term Interests Rates," Journal of Political Economy, 85, 661-89.

[5. Friedman, M. (1957) A Theory of the Consumption Function (Princeton: N.B.E.R.)

[6.] Gylfason, T. (1981) "Interest Rates, Inflation, and the Aggregate Consumption Function," Review of Economics and Statistics, 63-233-45.

[7.] Headon, R. S., and J. F. Lee (1974) "Life Insurance Demand and Household Portfolio Behavior," Journal of Risk and Insurance, 41, 685-98.

[8.] Hogan, J. D. (1970) "Life Insurance Company Investment Strategies: Lessons of the Sixties," American Risk and Insurance Association, mimeo.

[9.] Johnston, J. (1984) Econometric Methods, (New York: McGraw Hill).

[10.] Kindahl, J. K. (1963) "Saving in Life Insurance and Pension Funds: Some Problems of Economic Interpretation," Southern Economic Journal, 30, 25-39.

[11.] Linton, N. A. (1932) "Panics and Cash Values," Transactions of the Actuarial Society of America, 38, 365-94.

[12.] Modigliani, F. (1972) "The Dynamics of Portfolio Adjustment and the Flow of Savings through Financial Intermediaries," In Savings Deposits, Mortgages and Housing, E. H. Gramlich and D. M. Jaffe (editors), (Lexington, Massachusetts: D.C. Health).

[13.] Mohabbat, K. A., and E. O. Simos (1977) "Consumer Horizon: Further Evidence," Journal of Political Economy, 85-851-58.

[14.] Neumann, S. (1969) "Inflation and Savings through Life Insurance," Journal of Risk and Insurance, 36, 567-82.

[15.] Pesando, J. E. (1974) "The Interest-Sensitivity of the Flow of Funds through Life Insurance Companies: An Econometric Analysis," Journal of Finance, 49, 1105-21.

[16.] Revell, J. R. S. (1967) The Wealth of the Nation: The National Balance Sheet of the Unihted Kingdom (Cambridge: Cambridge University Press).

[17.] Schott, F. H. (1971) "Disintermediation through Policy Loans at Life Insurance Companies," Journal of Finance, 26, 719-29.

[18.] Weber, W. E. (1975) "Interest Rates, Inflation, and Consumer Expenditures," American Economic Review, 65, 843-58.

Introduction

Historical trends in most industrial countries indicate that, compared to other forms of financial saving, saving through life insurance has been declining over a long period. Explaining these trends requires examining the factors determining household saving through life insurance. In this connection, various hypotheses have been advanced in the literature, but few have been subject to rigorous empirical testing. (1) This article provides econometric tests of some of these hypotheses for the endowment policies written by British life insurers, using time series data for 1952 through 1985. Apart from these tests, the results also provide estimates of the degree to which saving through life insurance is sensitive to its various determinants.

The model of saving through life insurance used in this study is first outlined. Then empirical results are discussed and hypotheses about the relevance of various factors (e.g., interest rates) that have been suggested in the literature as exerting an important influence on such saving are tested. Households can adjust their saving through life insurance in a number of ways. Typically though, the main effects operate through changes in the allocation of new wealth to life policies and through surrenders. The empirical results based on the model estimated in this study do not, however, allow unambiguous inference about the effects on policy surrenders, which represent the main mechanism via which households adjust their existing holdings of the life insurance asset. Consequently additional evidence is provided on the hypotheses as they pertain to policy surrenders. These findings are, however, specific to one segment of the British life insurance sector: endowment policies. In conclusion, the findings are summarized and their implications discussed.

Saving through Life Insurance: The Model

Various options are available to holders of endowment policies written by British life insurers, in the event they wish to respond to changing capital market conditions by modifying their saving through these policies. They can simply forfeit their policies as they would term policies; they can make the policies paid-up, whereby, even though they cease making premium payments, they retain the benefits these policies offer; they can seek policy loans on the surrender value; or, they can seek a cash surrender.

Two hypotheses in the literature attempt to explain changes in household saving through life insurance, via the mechanisms mentioned above: (i) the interest rate hypothesis, and (ii) the emergency fund hypothesis. The former argues that saving through life insurance is sensitive to rates of return; positively to internal rates of return on endowment policies, and negatively to rates of return on other financial assets. The hypothesis has been examined with respect to policy loans in the United States by a number of researchers [see for example, Hogan (1970), Schott (1971), Pesando (1974), and Cummins (1978)]. The latter hypothesis, originally proposed by Linton (1932), is based on the premise that households regard their savings in life policies as a source of emergency funds which they can draw upon in times of need (e.g., during recession-induced unemployment), either by taking out policy loans or by seeking surrenders. Cummins (1975) tested this hypothesis for the United States for both policy loans and surrenders. His results did appear to support the hypothesis, particularly for surrenders during the 1950s. In the following section, British data on endowment policies are used to examine these hypotheses in terms of the sensitivity of saving through endowment policies to changes in their own rates of return and alternative rates of return, and in a variable constructed to reflect emergency fund considerations.

The Model

There is no unique, integrated theory of saving through life insurance, though several models of the demand for the life insurance product have been developed and used [e.g., Fortune (1973), Headon and Lee (1974), Kindahl (1963), Modigliani (1972), and Neumann (1969)]. There is also controversy surrounding the appropriate definition of saving through life insurance [see Cargill and Troxel (1979)]. The approach in this study is to develop an econometric model on the basis of the Modigliani (1972) stock-adjustment model, which in turn has been used and/or extended by others [e.g., Cummins (1975)]. With regard to the definition of saving, following Cummins (1975), saving through life policies are defined as the net flow of life insurance reserves. The Cummins measure excludes policy loans and examines them separately. The present measure is gross of policy loans, because unlike the general North American experience, policy loans from life insurers in Britain have been very small, and their exclusion in this study is likely to be inconsequential. It might be noted, in passing, that since a homogeneous segment of the life insurance sector is dealt with, the measure of saving is less likely to lead to biases inherent in more aggregated approaches.

The model adopted in this study is a variant of the Modigliani (1972) model and is based on Friedman's (1977) generalization of the basic Modigliani formulation. It can be stated as:

[Delta]A = [Beta] [Delta]W + [[gamma]W.sub.-1] [[Beta] - [b.sub.-1]]

or, in ratio form:

([Delta]A/[W.sub.1]) = [beta] ([Delta]W/[W.sub.-1]) + [gamma][[beta] - [b.sub.-1]

where

A = end-of-period stock of an asset W = end-of-period stock of household wealth b = end-of-period asset-to-wealth holding ratio [beta] = optimum asset-to-wealth holding ratio [gamma] = adjustment parameter (0 < [gamma] < 1) and the minus one subscript denotes a one-period lag.

According to this model, the flow of funds into an asset is made up of two parts: the first representing the allocation of new wealth to an asset according to the optimal holding ratio [beta], and the second reflecting a partial adjustment that involves a reallocation of existing wealth to that asset. The partial adjustment essentially implies that the allocation of existing wealth is more difficult than the allocation of new wealth. In the present context, [Delta]A can be interpreted as the net flow of funds into the life insurance asset; that is, as saving through life insurance. It is measured as the difference between the end-of-period and beginning-of-period stock of reserves. The reserves, as estimated by the Industrial Life Offices Association, are, however, not actuarial reserves, but simply the nominal value of life insurance funds. These funds are, therefore, not the theoretically correct measure of reserves. However, from an empirical point of view, this is not likely to be a serious problem, since the reserves are likely to be measured with error whether or not they are calculated as an actuary would calculate them. Thus the present measure of saving would be subject to measurement error in any case; and as is well known, measurement error in the dependent variable does not affect the quality of even the least squares estimator. (2)

In order to make equation (2) operational, the determinants of the optimal holding ratio [beta] must be specified. On the theoretical grounds, [beta] would, in general, be expected to be directly related to the internal rate of return for life insurance and inversely related to rates of return on competing assets. Whether this is borne out by the data is the central question concerning the validity of the interest rate hypothesis. In this study, instead of using a vector of alternative rates of return, only one such rate of return is considered to capture the effect of the whole spectrum of alternative rates of return.

The use of a single alternative rate of return would appear, at first glance, to be a major limitation. However, this is not the case. The aim is not to construct a consistent household sector portfolio model of asset choice since the objective is not to examine how various assets in a spectrum are related. Rather the focus is on the validity of the hypotheses mentioned above. Indeed, the emergency fund hypothesis need not imply inter-asset substitution at all. In fact, if that hypothesis is valid, it would have qualitatively similar implications for saving through all assets. Secondly, as far as the interest rate hypothesis is concerned, the focus is not on the issue of substitutability between the endowment policies written by the life insurers and each of the other assets. Instead, the study tests whether or not the flow of funds into an asset whose unique feature is saving cum protection is sensitive to market rates of return on other assets in general. For these reasons, the use of a single-alternative rate of return appears to be justified. Additional justification for this approach is provided by econometric considerations. As is well known, high collinearity leads to imprecise and unreliable estimates of parameters in the regression function. The inclusion of several market rates of return which tend to move together, would likely lead to these problems and render tests of the interest rate and emergency fund hypotheses unreliable. Therefore, a single rate of return is used to reflect returns on other assets that compete with the endowment policies.

The internal rate of return raises certain difficulties. First of all, there is no explicit rate of return offered on the life insurance asset considered in this article. There is undoubtedly an implicit yield, but there is no simple way in which that yield can be defined in an aggregate sense. Consequently, the usual approach is either to ignore it, or to use a market-determined interest rate to reflect the aggregate internal rate of return for a sector of the life insurance market. (3) This latter approach is somewhat suspect in that a proxy rate may well reflect the rate of return on competing assets, rather than the internal rate of return for endowment policies. Indeed, this is part and parcel of the collinearity problem alluded to above. Consequently, it would be extremely difficult to meaningfully interpret the interest rate effects when two (or more) such rates are included, one to reflect the rate of return on alternative assets, and one to act as proxy for the internal rate of return. (4) Seen in this light, the exclusion of a proxy rate may not appear to be an altogether inappropriate approach. (5) Though experiments were performed with this approach, the main results are based on a model that explicitly allows for the internal rate of return. This procedure, in fact, is such that the internal rate effects can be estimated without having to actually measure that rate. In this way, not only are collinearity problems circumvented, but also exclusion of a relevant variable with its attendant econometric problems is avoided. (6) With the foregoing discussion in mind, the determinants of the optimal asset holding ratio [beta] ban be specified.

The optimal ratio [beta], is specified as a linear function of the internal real rate of return i and a single real rate of return r reflecting the rate of return on other assets. That is,

[beta] = [[beta].sub.0] + [[beta].sub.1] r + [[beta].sub.2] i

On a priori grounds, the parameters are expected to satisfy the following restrictions: 0 [is less than or equal to] [[beta].sub.0] [is less than or equal to] 1; [[beta].sub.1] [is less than or equal to] 0; [[beta].sub.2] [is greater than or equal to] 0. Thus, if both r and i change by the same amount, then the optimal ratio [beta] will change if ~ [[beta].sub.1] ~ [is not equal to] [[beta]sub.2]. The direction and size of the change in [beta] depends on the size of each parameter, and cannot be predicted a priori.

Cummins argued that inflation could impact adversely on saving through life insurance, because of substitution from life insurance assets to physical assets, and to other saving media like inflation-indexed pension funds, which are better hedged against inflation. However, the effects of inflation on saving are not clear-cut, and there remains considerable uncertainty at both theoretical and empirical levels, about these effects. First of all, in terms of the formulation represented by equation (3), it is clear that if the expected rate of inflation rose, other things being equal, this would lower real estates. However, as stated above, the effect on saving of this equivalent change in real rates of return in ambiguous. The Cummins argument implicitly assumes, in terms of this formulation, that ~ [[beta].sub.] ~ [is greater than] [[beta].sub.2]. That is, the effect of inflation hypothesized by Cummins arises because of the differential impact of insurance's internal and alternative real rates of return on saving. The question that arises is whether inflation has any impact on [beta] that is not grounded in, and independent of, the real rates of return. Thus, if there is a change in expectations about inflation, real rates of return not chancing, would saving through life insurance change? In general, again, this cannot be determined on a priori grounds alone. Thus, if an increase in the expected rate of inflation increases the demand for nominal protection, other things being equal, saving through life insurance could increase of decrease. The answer depends on the distribution of life policies by type, since different policies require different amounts of reserves. In this study, however, only endowment policies are considered. These policies offer protection, but primarily provide an avenue for saving. With real rates constant, the effect of an increase in the expected rate of inflation depends on what happens to the demand for protection through endowment policies. It is possible that households would seek policies geared more protection, in which case saving through endowment policies would decline. However, this remains an empirical question.

On the basis of the foregoing discussion, it follows that:

[beta] = [[beta].sub.0] + [[beta].sub.1.r] + [[beta].sub.2.i] + [[beta].sub.3] [pi]*

where [pi]* is the expected rate of inflation.

Substituting equation (4) into equation (2) and manipulating yields:

([[Delta]A/W.sub.-1]) = [[[gamma]beta].sub.0] + [[beta'.sub.1.r] ([[Delta]W/W.sub.-1]) + [[beta].sub.2.i] ([[Delta]W/W.sub.-1]) + [[beta].sub.3.[pi]* ([[Delta]W/W.sub.1]) + [[[gamma]beta].sub.1.r] + [[[gamma]beta].sub.2.i] + [[gamma][beta].sub.3.][pi]* - [[gamma]b.sub.1] + [mu]X + [[beta].sub.0] ([[Delta]W/W.sub.-1])

where X represents the emergency fund hypothesis whose construction is discussed later.

Assuming that the real rates r and i can be expressed as the corresponding nominal rates r and I less the expected rate of inflation, equation (5) can be written in terms of the nominal rates as follow: (7)

([[Delta]A/W.sub.-1]) = [[[gamma]beta].sub.0] + [[beta].sub.0] ([[Delta]W/W.sub.-1]) + [[beta].sub.1.R] ([[Delta]W/W.sub.-1]) + [[beta].sub.2.I] ([[Delta]W/W.sub.-1]) - [[phi]pi]* ([[Delta]W/W.sub.-1]) + [[[gamma]beta].sub.1.R] + [[[gamma]beta].sub.2.I] - [[[gamma]phi]pi]* - [[gamma]b.sub.-1] + [mu]X

where [phi] = ([[beta].sub.1] + [[beta].sub.2] - [[beta].sub.3]).

In order to make equation (6) operational, three tasks must be completed. First, the nominal internal rate of return I must be eliminated from the equation. Second, how the expected rate of inflation is measured must be indicated. And finally, the emergency fund variable X must be defined.

As stated earlier, the major problem with life insurance's internal rate of return is that it is not an explicit, market-determined rate. However, whatever the implicit yield on endowment policies, market forces will ensure that that yield will be geared to market rates in general, even though evidence for the British life insurers indicates that this implic yield responds only sluggishly to market rates. The relationship between the internal rate I and market rates in general (as represented by R) can be approximated in statistical terms in several ways. Two representations considered in this article are as follows:

I = [[theta].sub.0] + [[theta].sub.1] [(R + [R.sub.1] + [R.sub.2])/3)] I = [[theta].sub.0] + [[theta].sub.1] [R.Sub.-1]

Both equations imply that the internal rate responds only sluggishly to market rates. Clearly, variations of this relationship, all more or less arbitrary, are possible. However, both specifications appeared to perform well, so the estimation is confined to them. If each of (7) and (8) is substituted into equation (6), manipulation yields:

([[DElta]A/W.sub.-1]) = [[gamma]delta] + [delta]([[Delta]W/W.sub.-1) + [[[beta].sub.1 + ([[beta].sub.2].[[theta].sub.1./3])] R ([[Delta]W/W.sub.-1]) + [[[beta].sub.2].[[theta].sub.1./3]] ([R.sub.-1] + [R.Sub.-2])([[Delta]W/W.sub.-1]) - [[phi]pi]* ([[DElta]W/W.sub.-1]) + [gamma][[[beta].sub.1] + ([[beta].sub.].[[theta].sub.1./3])] R + [gamma][[[beta]sub.2].[[theta].sub.1]./3] ([R.sub.-1] + [R.sub.-2]) - [[[gamma]phi]pi]* - [[gamma]b.sub.-1] + [mu] X

where [delta] = [[[beta].sub.0] + [[beta]sub.2].[[theta].sub.0]].

([[Delta]A/W.sub.-]) = [[gamma]delta] + [delta][[Delta]W/W.sub.-1]) + [[beta].sub.1.R] ([[Delta]W/W.sub.-1]) + [[beta].sub.2].[[theta].sub.1.([R.sub.-1])] ([[Delta]W/W.sub.-1]) - [[gamma]pi]* ([[Delta]W/W.sub.1]) + [[[gamma]beta].sub.1.R + [[[gamma]beta]sub.2].[[theta].sub.1].[R.sub.-1] - [[[gamma]phi]pi]* - [[gama]b.sub.-1] + [mu] X

Equations (9) corresponds to equation (7) for R, while equation (10) corresponds to equation (8) for R.

The emergency fund variable X is clearly a proxy variable, since there is no obvious or clear cut definition of what it should be. Two measures were considered, each of which is an unemployment-related construct and is therefore likely to be a reasonably good proxy for capturing the behavior implied by the emergency fund hypothesis. To be sure, other measures which reflect swings in the cyclical position of the economy could serve the same purpose.

The first measure is the annual rate of growth in the level of unemployment. This measure was used successfully by Cummins (1975). If the emergency fund hypothesis is valid, [mu] would be expected to be less than zero. That is, an increase in the rate of growth of unemployment, ceteris paribus, would reduce the net flow of funds into the life insurace asset, as households increased their policy loans and/or sought cash surrenders. With the second measure, X is defined as the level of actual unemployment relative to trend unemployment. Again, this measure would capture any sharp swings in economic activity, and if the emergency fund hypothesis is valid, [mu] would be expected to be less than zero.

Finally, a definition of the expected rate of inflation is considered. Expectations about inflation are assumed to be based on past rates of inflation. A popular model of expectations formation is the adaptive expectations model. (8) As is well known, if economic agents form their expectations of inflation adaptively, this is formally equivalent to defining [pi]* as a geometric weighted average of current and past rates of inflation. In other words:

[[pi].sub.t]* = [Mathematical Expression Omitted]

where [lambda], such that 0 [is less than] [lambda] [is less than] 1, is the weighting parameter. (9) Apart from considering this approach to expectation formation, a simpler definition of [pi]* was also considered. According to the latter definition, the expected rate of inflation was defined as a three-year arithmetic average of current and past rates of inflation. (10)

Estimation and Empirical Results

Several factors were considered in devising the estimation method. (11) To begin with, both equations (9) and (10) which represent the basic model, involve non-linear constraints on the coefficients of the variables, even though those equations are linear in variables. In order to take these restrictions into account (and this is necessary to get unique estimates of the parameters) the equations must be estimated by non-linear methods. (12) Second, it is immediately clear that some parameters in each equation are underidentified. That is, without further restrictions, only [[beta].sub.1] (the alternative rate of return effect) and [mu], the coefficient of the emergency fund variable can be estimated. Both parameters are of central interest in this study. However, [[beta.sub.2], the internal rate of return effect, and less importantly, [[beta.sub.3] and [[theta].sub.1] cannot be obtained. With the restriction: [phi] = [[beta].sub.1] + [[beta]sub.2] (that is, [[beta.sub.3] = 0), however, both [[beta].sub.2] and [[theta].sub.1] become estimatable. This restriction, which is also tested for, implies that inflation does not have an impact on saving through endowment policies that is indepdent of its impact operating through the real rates of return. These equations were estimated with and without this, and other testable restrictions.

The third factor which played an important role in determining the estimation method, was that since most of the right-hand side variables are likely to be endogenous and correlated with the disturbance term in the equations, the least squares estimators are biased, and the bias persists even in large samples. The instrumental variables estimation method was used to deal with this problem. (13) This method involves obtaining a set of variables which are uncorrelated with the disturbance term and using these as instruments for the endogenous variables. An added advantage of this method is that it also deals with the inconsistency of the estimates that results when explanatory variables are measured with error. (14)

Analysis of the Results

It is clear from the foregoing discussion that, given the two alternative definitions of both [pi]* and X, as well as the various parameter restrictions, these are several sets of estimates of eacch of equations (9) and (10). The results were very similar whether the geometric weighted average or the three-year arithmetic average definition of the expected rate of inflation was used. Further, the different definitions of the emergency fund variable X also gave very similar results. If anything, the rate of growth in unemployment definition of X produced equations that appeared to be somewhat more efficiently estimated. In light of this, the results reported below in Table 1 for equations (9) and (10) are based on the arithmetic average definition of [pi]*, and the rate of growth definition of X. (15)

The first two rows in Table 1 are the estimates of equations (9) and (10) with no restrictions. Both equations appear to fit the data well, and as a whole are highly significant at evven the 5 percent level. Further, tests on the Durbin-Watson statistic show that autocorrelation is absent from both equations. Turning to the parameter estimates, the estimate of [[beta]sub.1.], alternative rate of return parameter, not only has the right sign but is also significant at even the 1 percent level. This indicates that saving through endowment policies does respond to changes in market rates of interest. The effect of the internal rate of return [[beta].sub.2], on the other hand, cannot be estimated from these unrestricted equations. This is also true for the parameters [[beta].sub.3] and [[theta].sub.1]. Instead, only [[beta].sub.4] can be estimated, which is a composite of [[beta].sub.2] and [[theta].sub.1], and the results indicate that this estimate is highly insignificant. Estimates of [theta], the unrestricted coefficient of the inflation variable indicate that it is also highly insignificant. Further, there is no support at all for the emergency fund hypothesis, because the estimates of the parameter [mu] are statistically insignificant. The estimates of [gamma], the partial adjustment parameter, are barely significant at the 5 percent level and point to about a 2 percent elimination of the gap between the actual and optimal holding ratio in any period. This implies a slow and sluggish adjustment path to the optimal holding ratio.

In order to estimate the internal rate of return coefficient [[beta].sub.2], some restrictions must apply to the inflation coefficient [theta]. The logical restriction is: [theta] = [[beta].sub.1] _ [[beta].sub.2], or equivalently [[beta].sub.3] = 0. The estimates of equations (9) and (10) obtained with the restrictions, are presented in the third and fourth rows, respectively, of Table 1. In broad terms, the statistical properties of these equations are very similar to those obtained earlier. There is, in fact, only a slight change in the estimated coefficients. Of interest are the estimates of the coefficients of the interest rate variables. The estimate of the alternative rate of return effect [[beta].sub.1] is hardly changed in terms of both magnitude and statistical significance. In addition, estimates of the internal rate of return parameter [[beta].sub.2] are now available. These have the correct (positive) sign and are highly significant. Thus, the evidence suggests that both the alternative and internal rates of return do affect saving through endowment policies, and do so in the directions predicted by theory. However, again there is absolutely no support for the emergency fund hypothesis.

There are certain aspects of these results that merit further exploration. It may be noted that the results from the first pair of unrestricted equations support the restriction that the inflation coefficient [theta] = [[beta]sub.1] + [[beta].sub.2] - [[beta].sub.3] = 0. This in turn suggests one of the following restrictions: [[beta]sub.1] + [[beta]sub.2] = 0 was also tested and overwhelming support for it was found, thus ruling out the third possibility. This, with the implication (from the first pair of unrestricted equations) that [theta] = 0, supports the second set of restrictions. To explicitly test whether [[beta].sub.3] is indeed zero, equations (9) and (10) were estimated with the restriction that [theta] = -[[beta].sub.3], equivalently ([[beta].sub.1] + [[beta].sub.2]) = 0. These estimates are reported in the fifth and sixth rows of Table 1. Apart from some deterioration in fit, the results are very similar to those obtained without this restriction. More importantly, it is clear that the estimates of [[beta].sub.3] are highly insignificant, thus lending clear support to the restriction that [theta] = [[beta].sub.1] + [[beta]sub.2] = [[beta].sub.3] = 0. The estimates of equations (9) and (10) under this restriction are reported in the last two rows of Table 1. These estimates do not alter earlier conclusions about the hypotheses being tested. On the other hand, there appears to be some improvement in the efficiency of the estimates, as one might expect from the imposition of restrictions supported by the data. This improvement is reflected in reduction in the estimated standard errors of a number of estimated parameters. This final pair of equations also exhibits the best fit, and each is highly significant as a whole.

There are some additional aspects of the results that warrant further analysis. First of all, it has been shown that interest rates do matter in terms of their impact on saving through endowment policies. In order to gauge the strength of the interest rate effects, the elasticity of that saving can be obtained with respect to each rate of return. However, given that there are n o data for the internal rate, only the elasticity with respect to the alternative rate of return can be estimated. The expression for this elasticity ([xi]) is: (16)

[xi] = {[[beta].sub.1] [[Delta]W + [[gamma]W.sub.-1]]}R/[Delta]A

Based on the sample average values of the variables appearing in this expression, and on the average estimated values of [[beta].sub.1] and [gamma] in Table 1, the estimate of the elasticity is -1.4. This suggests that saving through endowment policies is interest elastic, that is, highly responsive to changes in alternative rates of return. It needs to be borne in mind, however, that this is a partial elasticity, in that the internal rate of return is not permitted to change when market rates change. Clearly, in practice, one would expect the internal rate to respond (sluggish though that response may be) to market rates of return, so that the total, as opposed to partial elasticity, would likely be smaller.

The second noteworth feature of these findings is the implication [[beta].sub.1] + [[beta].sub.2] = 0. This suggests that endowment policies are a one-for-one substitute with alternative financial assets, in that if both the internal and alternative rates were to change simultaneously by the same amount, there would be no net impact on saving through endowment policies. However, if the internal rate of return does not increase or increases only slowly when market rates increase, a substantial shift of saving from endowment policies to other assets is suggested by the high interest elasticity of such having.

When saving through endowment policies is measured as the net flow of funds into these policies, emergency fund considerations have not effect on such saving. This saving is, however, influenced by changes in the internal and alternative rates of return. Indeed, the elasticity estimate indicates a high responsiveness of saving to the alternative rate of return. Also, endowment policies and other assets appear to be one-for-one substitutes for one another, and inflation does not appear to have any net impact on saving through those policies.

Interest Rates, Emergency Fund Considerations and Policy Surrenders

One of the main insurance mechanisms through with changes in saving occur is a change in surrenders. Further, any change in surrenders will be reflected in a reversal in the net flow of life insurance saving. On the other hand, a change in the latter need not necessarily be associated with a change in surrenders. For instance, the net flow of funds can decrease without an increase in surrenders, if the new funds households wish to put into life insurance should decline for other reasons, give that there are alternatives to a straight surrender. For example, if households decide to forfeit policies or make them paid-up, the net flow would decline. Thus, it would appear these findings about the effects of interest rates and emergency fund considerations on saving through life insurance do not provide unambiguous information about the response of surrenders. In this context, it might be noted that the emergency fund hypothesis, if valid, would operate mainly by affecting surrenders. But this effect may not be captured by relating the net flow of funds into life insurance to the emergency fund variable, as in the previous section, because surreders are but one element in that net flow. It is therefore of some importance to empirically determine whether or not emergency fund considerations, or interest rates, have an independent impact on surreders.

In order to do this, the following model for surrenders is adopted:

S = [[alpha].sub.0] + [[alpha].sub.1.R] + [[alpha]sub.2.I] + [[theta]pi]* + [[alpha].sub.5.V.sub.-1] + [mu] X

where S = surrenders, V = end-of-period stock of life insurance funds, and X is the emergency fund variable measured in either of two ways described in the previous section. As before, R and I are, respectively, the alternative and internal rates of return. The stock of life insurance funds is included to capture the scale effect on surreders, and represents the upper bound on policy surrenders in any period. On a priori grounds one would expect that 0 [is less than or equal to] [[alpha].sub.5] [is less than or equal to] 1. One would also expect that [[alpha].sub.1] [is greater than or equal to] 0, and [[alpha].sub.2] [is less than or equal to ] 0, while [mu] would be positive if the emergency fund hypothesis valid, but zero otherwise. The sign of [theta], the coefficient of the inflation variable cannot be predicted on theoretical grounds alone. If real rates are assumed to be relevant to decision making, then according to the argument of the previous section, the effect of inflation on surrenders, via its impact on real rates, is given by - ([[alpha].sub.1] + [[alpha].sub.2]), and this sum cannot be signed on prior grounds. If inflation also has an impact on surreders that is independent of (in addition to) its impact via real rates, then:

[phi] = - ([[alpha].sub.1] + [[alpha].sub.2] - [[alpha].sub.3.])

where [[alpha]sub.3] is the impact of inflation, real rates of return remaining unchanged. This effect of inflation occurs if inflation has an impact on the demand for protection in general, and to the extent that endowment policies are not as protection-oriented as other policies, [[alpha].sub.3] might be expected to be positive. In any event, it is clear that the unrestricted effect of inflation, as measured by [theta], cannot be signed on a priori grounds. Nor can the parameters [[alpha].sub.2] and [[alpha]sub.3] be identified or estimated.

In order to make equation (11) operational, the internal rate of return variable I must be eliminated. This is done following the procedure outlined in the previous section. Thus, using equation (7) for R, it can be shown that the equation (11) reduces to:

S = [delta] + ([[alpha]sub.1] + [[alpha.sub.4]) R + [[alpha].sub.4] ([R.sub.-1] + [R.sub.-2]) + [[phi]pi]* + [[alpha].sub.5] [V.sub.-1] + [mu] X where [delta] = [[[alpha].sub.0] + [[alpha].sub.2].[[theta].sub.0]], [[alpha].sub.4] = ([[alpha].sub.2].[[theta].sub.1./3]), and [theta] = - ([[alpha].sub.1] + [[alpha].sub.2] - [[alpha].sub.3]). On the other hand, if the approximation is used for I given by equation (8) the estimating equation becomes:

S = [delta] + [[alpha].sub.1.R + [[alpha].sub.4.R.sub.-1] + [[theta]pi]* + [[alpha].sub.5] [V.sub.-1] + [mu] X

A number of points about the estimation of these equations may be noted. First, only equation (12) requires non-linear estimation. Equation (13) has no non-linear restrictions, and can thus be estimated by linear methods. Second, [[alpha].sub.2], [[alpha].sub.3] and [[theta].sub.1] are not identified, and thus cannot be estimated. Finally, each of these equations was estimated using the two definitions of the emergency fund variable, as well as the two formulations of the expected rate of inflation.

In general how expected inflation is measured is not important in terms of the nature of the findings. Different definitions of the emergency fund variable X also gave broadly similar results, with the same general conclusions. One major difference was that while the actual-to-trend unemployment definition of X led to better statistical fits than the annual growth rate of unemployment definition, the former performed rather poorly in terms of autocorrelation even when a correction for it was possible. The latter definition, on the other hand, gave results that did not suffer from this problem. As a consequence, the latter results are likely to be more reliable. These results are reported in the first two rows of Table 2.

It may be n oted that the estimates of equation (12) have been obtained after adjusting for autocorrelation using the Corchrane-Orcutt method. The (adjusted) coefficient of determination declines sharply. However, this statistic is neither comparable to its value in other equations, nor is it very meaningful in the present context. (17) More importantly, it is clear from the Durbin-Watson statistic that neither equation suffers from autocorrelation.

It is clear that only the estimates of the coefficients of the stock variable and the emergency fund variable are statistically significant at the 5 percent level. The estimate of the alternative rate of return effect, as measured by [[alpha].sub.1], is highly insignificant in the first equation, and just barely so in the second. In both equations, this coefficient has the wrong sign, and it is likely that is has not been well estimated in either. The same applies to the estimates of the composite parameter [[alpha].sub.4], as well as the inflation parameter [theta], both of which are highly insignificant. In short, the estimates of equations (12) and (13) provide support for the emergency fund hypothesis, but neither the alternative rate of return nor inflation appear to have any impact on surrenders.

In the versions of the models estimated above, it was not possible to estimate the internal rate of return effect as measured by [[alpha].sub.2], since that parameter is not identified. In order to estimate the internal rate effect, the identifying restriction [[alpha].sub.3] = 0 can be imposed. Under this restriction, [[alpha].sub.1] and [[alpha].sub.2] as well as [[theta].sub.1] can be estimated, and the inflation coefficient [theta] is estimated by the sum: - ([[alpha].sub.1] + [[alpha].sub.2]). However, estimation of the parameters of both equations (12) and (13) requires non-linear methods. The results for these equations, under the afore-mentioned restriction are reported in the last two rows of Table 2.

An examination of the estimates shows that results are quite similar to those obtained earlier. The alternative rate of return effect has the wrong sign, but is highly insignificant in both equations. Again, the stock variable has a significant impact on surrenders, and the emergency fund hypothesis is also supported, as the estimate of [mu] is significant at the 5 percent level. The estimates of the two new parameters, [[alpha].sub.2] and [[theta].sub.1], are both highly insignificant. Thus, surrenders in British endowments do not appear to be affected by interest rates.

It is interesting to note that the above findings on the impacct of interest rates, and on the emergency fund hypothesis are just the opposite of those obtained in the previous section. This suggests that while saving through endowment policies is affected by interest rates, the adjustment takes place through channels other than policy surrenders. This implies that interest rates typically determine how households allocate their new wealth between assets but do not affect existing wealth in the form of endowment policies through surrenders. On the other hand, emergency fund considerations affect saving through endowment policies in just the opposite way.

Conclusions

Empirical findings suggest that saving through endowment policies are responsive to internal returns for British endowments and to alternate rates of return on alternative investments. Furthermore, if both internal and alternative real rates change by equivalent amounts, there is no impact on saving through those policies, thus implying that those policies and other financial assets are, in general, one-for-one substitutes. On the other hand, since intrenal rates respond only sluggish to changes in market rates, one would expect a strong adverse impact on saving through endowment policies if market rates were to increase. (18) Indeed, the declining long-term trend in saving through endowment policies relative to saving in other assets, may well reflect the sluggish response of the internal rate for endowments to generally rising market rates. The interest sensitivity of saving, however, cannot be attributed, even in part, to surrenders. In fact, no significant relationship between surrenders and rates of return was found. Thus, it appears that changes in rates of return affect saving through endowment policies primarily through their impact on the flow of new wealth.

Emergency fund considerations were found to have no influence on saving through endowment policies. On the other hand, surrenders were found to be significantly related to the emergency fund variable. However, since the emergency fund hypothesis is primarily a cyclical phenomenon, these findings imply that emergency fund considerations have a short-run impact on saving through endowment policies, but cannot explain the observed trends relative to other forms of saving. Findings on the rate of return effects appear to be consistent with those trends.

While this study deals only with the endowment policies of British life insurers, findings that saving through those policies are highly sensitive to relative rates of return, and (via surrenders) to emergency fund considerations, may well apply to other segments of the life insurance sector. In that event, the life insurance sector as a whole, could be subject to the type of short and long term effects found for endowment policies. Thus, a period of relatively rising alternative rates of return could sharply shift saving patterns in favor of assets offered by other financial intermediaries. This would imply a shift in the volume and composition of capital flows into bond and equity markets, with consequent wider macroeconomic effects. On the other hand, findings for endowment policies written by British life insurers also point to important distributional effects within the life insurance sector. For example, a shift in saving away from endowment policies (as a result of a change in rates of return) may reflect policyholders' desire to separate protection from saving. This could result in a movement towards protection-only policies, and since any compensating increase in such policies does not generate the same amount of reserves for investment, there could be a decline in the aggregate flow of investible funds.

In conclusion, as is the case in most empirical studies, some caution in warranted in using quantitative results to make inferences. In this context, these results are specific to the endowment policies of British life insurers. Thus, any inferences about the implications of these findings for the life insurance sector as a whole are merely suggestive. On the other hand, they appear to be plausible, and are consistent with observed trends within the British life insurance sector.

Appendix: Variable Definitions and Data Sources

Wealth (W) is defined as the net worth of the personal sector and is calculated as follows: A benchmark estimate of W for 1957 was taken from Revell (1967), p. 163, Table 7.6, and cumulated through 1985 with the help of Britain's personal saving series. The latter data were drawn from various issues of the Blue Book -- National Income and Expenditure, HMSO.

The data for surrenders (S) and for the stock of life insurance reserves (A) were from the Industrial Life Offices' Association (ILOA). Saving through endowment policies were then measured as the difference between end-of-period reserves and beginning-of-period reserves, while b, was measured as: A/W. It might be noted that the estimates of life insurance reserves published by the ILOA are not actuarial reserves but reflect the market value of life insurance funds.

For the rate of return on alternatives assets, the 2.5 percent of War loan (a perpetuity) end-of-year flat yield was chosen. The data for this wate were taken from various issues of the Bank of England Quarterly Bulletin.

The inflation rate ([pi]) was measured as the annual percentage change in the consumer price index, data for which were obtained from issues of International Financial Statistics: Yearbook, the International Monetary Fund.

The instrumental variables chosen (other than those variables that appear in the estimating equations) were the following: (1) exports ([pound sterling] million) (2) unit value index for exports (1975 = 100) (3) foreign exchange reserves (beginning of period -- $ million U.S.) (4) government expenditure ([pound sterling] billion) (5) government consumption ([pound sterling] billion) (6) mid-year population (millions). The data for these were drawn from various issues of International Financial Statistics: Yearbook, The International Monetary Fund.

(1) See, however, Cummins (1975) and Pesando (1974).

(2) See Johnston (1984), pp. 428-30. See also the appendix for data resources, and for definitions of all other variables.

(3) Modigliani (1972) uses the mortgage rate, while Cummins (1975) uses the rate on directly placed bonds in the United States.

(4) The upshot of this is that some market-determined rate of return would better reflect movements in the rates of return on alternative assets, than it would changes in the implicit return on the life insurance asset. This is because alternative rates of return are themselves market rates.

(5) On the other hand, the exclusion of a relevant variable could lead to biased and inconsistent estimates, if the estimation method is not appropriately amended. See Johnson (1984), pp. 259-64. This leads to a rather serious dilemma in that the inclusion of a variable when it is highly collinear with other included variables gives rise to unreliable estimates. It is generally more difficult to deal with the latter problem.

(6) In fact, this estimation method is sufficiently robust to circumvent these econometric problems as well.

(7) Alternatively, R = r + [pi]*, and I = i + [pi]*.

(8) Versions of this formulation have been used extensively is saving/consumption studies. See, for instance, Weber (1975) and Gylfason (1981). If expectations about inflation were adaptive, they could be represented as follow:

[pi]* = [lambda] + (1 - [lambda])[pi]* -1.

where [lambda], such that 0 [is not equal to] 1, is the coefficient of expectation.

(9) This equivalence can easily be shown by using this definition in the equation, and then applying a Koyck transformation. See Johnston (1984), pp. 346-48.

(10) This is undoubtedly a simple formulation, but it has been used with success in savings studies. Some support for using three years comes from the finding that the period adequately defines the time horizon of households [see Friedman (1957), Mohabbat and Simons (1977)].

(11) It has been implicitly assumed that each equation is subject to a disturbance term.

(12) The non-linearity arises because the coefficients are non-linear combinations of each other. Of course, if one were not interested in the individual parameters, the equations could be estimated by linear methods.

(13) See Johnston (1984), pp. 363-66, and 430-32, for this method of estimation. A careful choice of instruments would generally require knowledge of the set of pre-determined variables in a larger model of the British economy. A set of variables has been chosen that is likely to satisfy this requirement. A part of this set is made up of the explanatory variables. The rest of the variables do not explicitly appear in the equations, but are viewed as pre-determined within the context of a larger model, of which this one is a part. These variables are listed in the appendix.

(14) The implications of measurement error are similar to those resulting from the exclusion of a relevant explanatory variables. both lead to the general problem of correlation between the explanatory variables and the disturbance term. See Johnston (1984), pp. 428-35.

(15) All estimation was done using the Time Series Processor (TSP), Version 4.0 computer package.

(16) This elasticity is defined in the following way: [xi] (* log [Delta]A/* log R) = (* [Delta]A/* R) (R/[Delta]A), where the bar denotes the sample mean of the variable concerned.

(17) In this equation, the correction for autocorrelation involves a transformation of the equation, so that the dependent variable is no longer the same. As a consequence, the coefficient of determination is not directly comparable across equations. In addition, no clear meaning can be attached to any measure of goodness of fit, since the transformation is predicated only on econometric, and not theoretical grounds. It is also uncommon for the fit of the transformed model to deteriorate.

(18) The strong impact results from the high interest sensitivity of saving.

A. Dar is Associate Professor and J. C. Dodds is Dean of Commerce at Saint Mary's University, Halifax. An earlier draft of the paper benefited from comments from two anonymous referees and from Dr. S. Amirkhalkhali, however, the authors accept responsibility for any errors.

References

[1.] Cargill, T. F., and T. E. Troxel (1979) "Modeling Life Insurance Savings; Some Methodological Issues," Journal of Risk and Insurance, 46, 391-410.

[2.] Cummins, J. D. (1975) An Econometric Model of the Life Insurance Sector of the U. S. Economy (Lexington, Massachusetts: D.C. Health).

[3.] Fortune, P. (1972) "A Theory of Optimal Life Insurance: Development and Tests," Journal of Finance, 28, 317-26.

[4.] Friedman, B. J. (1977) "Financial Flow Variables and Short-Run Determination of Long-Term Interests Rates," Journal of Political Economy, 85, 661-89.

[5. Friedman, M. (1957) A Theory of the Consumption Function (Princeton: N.B.E.R.)

[6.] Gylfason, T. (1981) "Interest Rates, Inflation, and the Aggregate Consumption Function," Review of Economics and Statistics, 63-233-45.

[7.] Headon, R. S., and J. F. Lee (1974) "Life Insurance Demand and Household Portfolio Behavior," Journal of Risk and Insurance, 41, 685-98.

[8.] Hogan, J. D. (1970) "Life Insurance Company Investment Strategies: Lessons of the Sixties," American Risk and Insurance Association, mimeo.

[9.] Johnston, J. (1984) Econometric Methods, (New York: McGraw Hill).

[10.] Kindahl, J. K. (1963) "Saving in Life Insurance and Pension Funds: Some Problems of Economic Interpretation," Southern Economic Journal, 30, 25-39.

[11.] Linton, N. A. (1932) "Panics and Cash Values," Transactions of the Actuarial Society of America, 38, 365-94.

[12.] Modigliani, F. (1972) "The Dynamics of Portfolio Adjustment and the Flow of Savings through Financial Intermediaries," In Savings Deposits, Mortgages and Housing, E. H. Gramlich and D. M. Jaffe (editors), (Lexington, Massachusetts: D.C. Health).

[13.] Mohabbat, K. A., and E. O. Simos (1977) "Consumer Horizon: Further Evidence," Journal of Political Economy, 85-851-58.

[14.] Neumann, S. (1969) "Inflation and Savings through Life Insurance," Journal of Risk and Insurance, 36, 567-82.

[15.] Pesando, J. E. (1974) "The Interest-Sensitivity of the Flow of Funds through Life Insurance Companies: An Econometric Analysis," Journal of Finance, 49, 1105-21.

[16.] Revell, J. R. S. (1967) The Wealth of the Nation: The National Balance Sheet of the Unihted Kingdom (Cambridge: Cambridge University Press).

[17.] Schott, F. H. (1971) "Disintermediation through Policy Loans at Life Insurance Companies," Journal of Finance, 26, 719-29.

[18.] Weber, W. E. (1975) "Interest Rates, Inflation, and Consumer Expenditures," American Economic Review, 65, 843-58.

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Author: | Dar, A.; Dodds, C. |
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Publication: | Journal of Risk and Insurance |

Date: | Sep 1, 1989 |

Words: | 8166 |

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