# Whole life cost comparisons based upon the year of required protection.

Whole Life Cost Comparisons Based Upon the Year of Required
Protection

ABSTRACT

The traditional measures of Interest Adjusted Surrender Cost and Linton's rate of

return are computed for 68 whole life policies. Similar measures are obtained via linear

programming where cash flows are discounted using several different external interest

rates. With the linear programming method, year of insurance protection is varied to

include year 0, year 10 and year 20. If protection is required in year 20, several policies

are infeasible (incapable of generating any insurance protection). Upper limits for rates

of return are calculated. Infeasibility occurs for a given policy if the external rate of

return exceeds the upper limit.

Introduction

The purpose of this article is to use a linear programming (LP) method for measuring the cost of whole life insurance. The method is applied to policies offered in 1984 by 68 different insurers. For comparative purposes, the traditional methods of interest adjusted surrender cost (IASC) and Linton's rate of return are applied to the same set of policies. The insurers and policies were selected using criteria established in an earlier study by Hutchins and Quenneville [3]. Consequently, the results of this study of 1984 policies may be compared to a set of similar policies offered in 1972.

The LP method is similar to both IASC and Linton methods in several respects. Specifically, all three methods assume deterministic projected dividends and a fixed horizon of 20 years. The LP technique can be used to derive level interest adjusted costs similar to the IASC and internal rates of return on equity similar to the Linton rate of return. Nevertheless, both the IASC method and Linton's method rely on assumptions not required by the LP method. The LP method requires only an assumption of a rate of return that is relevant to the policyholder.

Conceptually, the IASC and Linton's method treat the whole life insurance policy as providing the two products of protection and savings. The IASC attempts to measure the level cost of protection while the Linton method attempts to measure the rate of return on the savings component. The IASC requires an interest rate for discounting whereas Linton's method requires various term insurance rates. As a consequence of these different required assumptions, rankings obtained by these two methods will not be perfectly correlated.

The LP method, by contrast, does not attempt to directly separate protection and savings. The method assumes that the insured individual requires a given level of protection. It is irrelevant to the insured how the insurer provides the protection. In other words, the insured is unconcerned with how the insurer divides the premium into loading charges, reserves, and so forth. The LP method has the additional flexibility of considering the point in time at which the insured requires protection. Neither the IASC nor Linton method explicitly address issues related to timing of insurance protection.

The flexibility of varying the year of required protection is the primary characteristic of the LP method that differentiates it from the traditional methods. The IASC, for example, implicitly assumes that coverage is required at the time the policy becomes effective. The IASC, however, does not recognize the reduction in the insured's wealth resulting from the premium payment. The LP method recognizes premium payments by increasing the policy face value by a corresponding amount. With the LP method, premiums are compounded and accumulated at the relevant interest rate to the year in which protection is required. This accumulation of premiums may be large if protection is required 15 or 20 years into the future. For large interest rates the accumulated premiums will, at some point, exceed the face value of the policy. The upper bound interest rate at which accumulated premiums equal exactly the face value of the policy will be presented below for the various policies studied.

I. Measures of Whole Life Insurance Cost

The interest adjusted surrender cost, recommended by the Joint Special Committee on Life Insurance [6], is computed by discounting each annual premium, per $1000 face value, by an appropriate interest rate. Premiums are reduced appropriately to reflect projected dividends, if applicable. The cash value at the horizon plus any terminal dividend is discounted and subtracted from the sum of discounted premiums. The resulting difference is divided by the annuity factor corresponding to the interest rate. The end result provides a measure of the level cost per $1000 of insurance.

A major problem with the IASC is the choice of an interest rate for discounting. Ideally the interest rate should reflect the return that the insured could realize on an investment similar to the equity inherent in the policy. The choice of interest rate should be related to the uncertainty of cash flows. The largest amount of uncertainty occurs from dividend payouts, since actual dividend payouts seldom match precisely the projected dividends. In other words, the uncertainty of dividend payouts introduces an element of risk into the policyholder's portfolio. Also, nominal interest rates vary considerably over time so that a rate used ten or 15 years ago may not be appropriate today. Yet little attempt has been made to adjust the rate used to compute IASC's reported in the trade periodicals. The only observable change is that, in recent years, IASC's have been computed using an interest rate of 5 percent rather than 4 percent. Clearly, even the risk free rate has exceeded 5 percent in recent years. As a result, the IASC's for many policies are negative. In addition to being intuitively unappealing, negative IASC's may be misleading. Because of the structure of premiums, dividends and ending cash values, the rankings of a given set of policies may differ when different interest rates are used.

Another problem associated with the IASC is that the amount of insurance protection is based upon the face value of the policy. Nevertheless, the effective amount of insurance protection inherent in a policy is actually less than the face value if the cash outlay of the premium is taken into consideration. By paying the premium, the insured has reduced his or her asset level and thereby his or her capacity to self insure. By ignoring this factor, the IASC of a policy can be decreased arbitrarily, for example, by increasing the premium and making appropriate increases in the horizon cash value.

A problem related to the effective amount of protection generated by a policy is the timing of protection required by the insured. Typically the horizon length for analysis and measurement purposes is either ten or 20 years. Setting the horizon at 20 years, however, does not guarantee that the policy under consideration is providing the face value of protection at the horizon. To obtain the precise amount of effective protection in a given year, the face value amount must be adjusted to reflect the accumulated premiums paid up to that year. It is possible, as demonstrated in the latter part of this article, that the premiums accumulated at interest exceed the face value of the policy. In essence, such policies provide no effective protection.

In contrast to the IASC, which measures the cost of protection given a rate of return, Linton's method measures the policy rate of return given the cost of protection. Although not commonly reported in trade publications, Linton's rate of return method is well known and documented in the literature. With this method, the actual insurance cost is separated from that part of the premium designated for policy equity. The computations are somewhat involved, but in the end an internal rate of return is obtained. The reader is referred to the original work of Linton [5] and an article by Belth [1] for details on computation of the rate of return. The derived rate of return provides a measure that can be compared to rates of return on other financial instruments. Nevertheless, some problems remain with the rate of return method. For example, the amount of pure insurance protection considered in an analysis varies as the trial rate of interest varies. The major problem with this method, however, is that it is necessary to assume term rates so that the insurance provided can be separated from protection provided by the policy equity. If high term rates are used, then the resulting rate of return will be higher, whereas if low term rates are used, then the resulting rate of return will be lower. How does one choose appropriate term rates for such an analysis? No one has satisfactorily answered this question. For example, Hutchins and Quenneville [3], who fully recognized this problem, used composite term rates. These composite term rates were computed by averaging the term rates for several insurers. As with the IASC, the rate of return method does not address the timing of protection.

The third method considered in this article, the linear programming method, imposes constraints for the insured and maximizes the discounted cash flows associated with a given policy. Two types of constraints are present. First, constraints are included to represent amounts that the insured is willing to budget for insurance. For purposes of developing comparative measures, these amounts are set to zero. Since the linear programming model includes variables for borrowing and lending, setting the budget amounts to zero forces the solution to include positive amounts for borrowing. The second set of constraints establishes limits on the amounts of protection required in each year to the horizon. For purposes of developing a cost measure, the analysis that follows includes only a single protection constraint corresponding to the year in which insurance is required. A single variable is used to represent the face value of the policy, purchased at the outset, necessary to satisfy all of the constraints. The complete mathematical linear programming formulation is included in the Appendix.

In effect, the LP method measures the ability of a given policy to provide a specified level of protection for a given number of years. That is, the LP method is concerned only with a specified level of protection rather than how the protection and savings components are separated within a given policy. The horizon length, however, remains the same as with the traditional methods (e.g., 20 years). With the LP method also, it is necessary to assume an interest rate for discounting. The interest rate used, however, is an external rate at which the insured can borrow and lend. Although different rates could be used for borrowing and lending, for simplicity it is assumed that these rates are identical. Since the budget amounts are equated to zero, only the borrowing rate is relevant. The negative of the optimal objective function value is the discounted cost of the policy over the 20 year horizon. Dividing this objective function value by the annuity factor corresponding to the external rate yields a level interest adjusted cost. An identical interest adjusted cost is obtained by observing the dual price corresponding to the single insurance constraint. This dual price can be interpreted as the discounted marginal cost of an additional dollar of insurance protection. Multiplying by 1000 and dividing by the 20 year annuity factor produces the level interest adjusted cost per $1000. The full derivation is provided in Schleef [8].

In addition to the insurance protection dual price, the LP method also uses the dual prices for the budget constraints. The dual prices corresponding to each of these constraints can be interpreted as a marginal discount factors. That is, the dual price for the budget constraint in year t denotes the present value of having one additional dollar available in year t. By summing these dual discount factors for each year to the horizon, an annuity factor is obtained. Finding the corresponding level interest rate for this annuity factor provides a marginal rate of return for the policy. Consequently, the LP model yields jointly an IASC and a rate of return, each of which is relative to the assumed external rate of return.

II. Characteristics of The LP Method

The precise inverse relationship between level cost of insurance and marginal rate of return is given in [8]. The level interest adjusted cost obtained with the LP method is similar to the traditional IASC. For example, if the protection requirement constraint for the first year is set equal to $1000, then the IASC derived by the LP method will differ from the traditional IASC by a constant factor. To demonstrate this point, if the first year premium is $18 per $1000 face value, then the LP model will call for $1018 of face value insurance. The traditional IASC will, in this case, equal 0.9823 times the LP IASC, i.e., 0.9823 = 1000/1018. The linear programming method, however, allows the flexibility of requiring a given level of protection in years other than the first. By setting the budget constraints to zero, the effective amount of protection generated is measured rather than the face value amount of the policy. As the year in which insurance is required occurs farther into the future, the corresponding level cost of insurance increases. This happens because the model requires that the initial policy is sufficiently large to cover the premiums up to the year in which the protection is required. In addition, the premiums are compounded at the (external) interest rate used. This effect will be demonstrated later in the paper for a selected policy.

The marginal rate of return obtained from the LP method is relative to the external rate used for borrowing and at which the cash flows are discounted. This marginal rate of return does not measure directly the rate of return of the equity component of the policy, but rather provides a marginal rate of return on the policyholder's wealth relative to the policy. It can be shown that the marginal rate of return derived from the LP method will never exceed the external rate of return used for borrowing and discounting. The difference between the marginal rate of return and the external rate represents the loss, in terms of rate of return, that the insured incurs as a result of holding the insurance policy.

Each policy has two limiting rates of return related to the LP method that warrant discussion. Both of these limits apply to the external rate of return used for discounting. The lower limit rate of return corresponds to the external rate of return at which the LP solution becomes unbounded. That is, for external rates of return below this limit, the objective function increases without bound. The interest rate at which this occurs satisfies the following equation where [PLD.sub.t] denotes the premium less dividend for year t, [CV.sub.n] denotes the cash value plus any dividends at the horizon, and i denotes the interest rate.

[Mathematical Expression Omitted] The limiting rate of return, i, is the internal rate of return that occurs if the cash flows associated with the policy are evaluated without any consideration for the insurance inherent in the policy. As the external rate of return approaches this lower limit, the marginal rate of return derived from the dual LP discount factors also approaches this lower limit. The lower limit for rate of return is the internal rate of return realized if only the equity component of the policy is considered independent of the face value insurance provided in the policy.

The upper limit on the external rate of return represents that rate of return at which the linear programming solution is infeasible. Infeasibility implies that the policy under consideration is incapable of providing any actual insurance protection. Such infeasibility occurs if the premiums, compounded at the external rate of return, exceed the face value of the policy. For example, if insurance is required for the 20th year, infeasibility occurs if the compounded sum of the premiums exceeds the face value of the policy. Mathematically infeasibility occurs if

[Mathematical Expression Omitted] where j denotes the latest year protection is required.

Since the first year premium never exceeds $1000 per thousand dollars of face value, infeasibility never occurs when the insurance requirement occurs for the first year. As the insurance requirement occurs farther into the future, the external rate of return at which infeasibility occurs decreases.

III. Application of The Linear Programming Model

In this section, the LP method is used to compute comparative measures for 68 whole life policies. The results of the LP analysis are compared to results obtained by using the traditional measures of IASC and Linton rate of return.

The 81 insurers considered by Hutchins and Quenneville [3] were used as a basis for selecting policies for analysis. The data used in the present study were collected from Best's Flitcraft Compend 1984 [2].(1) The criteria used for selection were also similar to those used by Hutchins and Quenneville (henceforth referred to as H & Q). The present study considers policies of $25,000 for males age 35 compared to policies with a face value of $10,000 considered by H & Q. Of the 81 insurers considered by H & Q in 1972, 68 reported policy data in Best's Flitcraft Compend 1984 [2] meeting the various criteria. For several, minor differences from those policies considered by H & Q may exist. For example, 36 of the 68 policies analyzed below are for nonsmokers. In order to compute Linton's rate of return, term rates are required. Data from ten non-participating five- year renewable term policies were collected. Seven of the term policies were standard policies while the remaining three policies were for nonsmokers. The rates reported for males age 35, 40, 45, and 50 were averaged separately to obtain the following term rates. These rates are used for the age given and the four subsequent renewal years.

To provide a basis for comparison with the H & Q study, Table 1 lists the rate of return and IASCs for each insurer. The IASCs are computed using interest rates of 4, 5, 8, 10, and 12 percent. H & Q used an interest rate of 5 percent--the most commonly used rate in the industry--in their study. Note that IASCs reported in Table 1 are computed using projected dividends for policies offered in 1984 whereas, H & Q also examined the actual historical performance of policies.

One of the main results of the H & Q study was the large negative correlation between IASC and Linton rate of return. Ranking the 68 policies on these two measures yields a rank order correlation coefficient of -.938 compared to -.979 for the H & Q study. Substantial differences in the levels of each of these two measures exist between the two studies. For example, the term rates used in the present study are approximately one third less than the corresponding rates used by H & Q. Nevertheless, the mean Linton rate of return is 7.82 percent with a standard deviation of 2.02 compared to a mean of 4.75 percent and standard deviation of 0.73 reported by H & Q for policies issued in 1972. Using an interest rate of 5 percent yields a mean IASC of $1.44 per thousand with a standard deviation of 2.44 for the 68 insurers in this study. This compares with a mean of $6.01 and standard deviation of 1.07 in the H & Q study.

As reported above, the 1984 Linton rates of return are higher and the IASCs are lower than in the H & Q study. Moreover, Table 1 demonstrates that different rankings of insurers occur as the interest rate used for discounting is changed. One of the conceptual problems with the IASC method is choosing an appropriate rate for discounting. Given the default free rates prevailing in 1984, such as T-bond rates, a rate of 5 percent is too low. If the IASC in some sense measures cost of insurance, then negative IASCs suggest negative costs of insurance. To the extent that insurers and insureds compete in the same investment market, it is unlikely that life insurance will have a negative cost. Consequently, a more reasonable interest rate for computing IASCs should, at a minimum, yield positive IASCs. For example, Table 1 indicates that interest rates of 8 percent and above yield positive IASCs for each insurer.

The various cost measurement methods discussed in this article will be demonstrated by an example. Exhibit 1 provides such an example. The policy data pertain to a $25,000 policy issued by Connecticut Mutual Life. The IASCs computed by the traditional method indicate that a policy's ranking depends upon the interest rate used for discounting. For example, the given policy has the smallest IASC if a 4 percent rate is used, but the policy ranks 35th out of 68 if a rate of 12 percent is used. The policy is ranked 25th (tie) for the Linton return. The linear programming measures yield rankings similar to the traditional method, given that small interest rates are used. This is not surprising, especially for the case of requiring protection at the outset (year 0). In the year 0 protection case the LP IASC differs from the traditional IASC by a constant. That is,

LP IASC = Trad. IASC/(1-0.02155) where the cost per $1000 of face value (protection) is $21.55. In effect, the LP measure requires additional insurance to cover the initial premium. In this example, it is necessary to purchase $1022.02 of face value in order to generate $1000 of protection, i.e.,

$1022.02 = 1000/(1-0.02155) For those cases in which protection is required in later years, such as years 10 or 20, the LP IASC and the traditional IASC also differ by a constant. But in these cases the constant accounts for insuring against the compounded premiums paid up to and including the year of protection. Schleef[8] provides the formula to compute the LP IASC for the general case.

Even though the traditional IASCs and the LP IASCs differ by a constant for each policy, the policy rankings for these two methods are not identical. For example, with an interest rate of 12 percent, the traditional method results in a ranking of 35 for the Connecticut Mutual policy. With the LP method, using an external interest rate of 12 percent yields a ranking of 36 for year 0 protection, but a ranking of 40 if protection is required in year 10. Moreover, if protection is required in year 20, no feasible solution exists. This means that the sum of the first 20 premiums per $1000 protection, adjusted for projected dividends and compounded at 12 percent, exceeds $1000 by the beginning of year 20. Feasibility occurs only if the interest rate is less than the upperbound of 11.74 percent (see Table 4). The upperbound limit of 11.74 percent ranks this policy 51st in terms of the upperbound rate, given protection is required in year 20. In contrast, the lower bound interest rate of 6.14 percent is relatively high with a ranking of 4. This means that the cash flows resulting from premiums, projected dividends, and terminal cash value will yield 6.14 percent over a 20 year period. To summarize, the Connecticut Mutual policy ranks first in terms of the traditional IASC, given the low interest rates of 4 and 5 percent, but is incapable of providing any actual insurance protection if the interest rate is 12 percent and protection is required in year 20.

The LP method of analysis requires that an external interest rate be specified. This external rate represents the insured's opportunity cost of foregoing investment so that insurance may be purchased. The second factor included in the linear programming analysis is the year in which insurance protection is required. With the Linton and IASC methods the face value protection occurs at the outset, but there is no guarantee as to the level of insurance protection in subsequent years. If the amount of the premium must also be covered, then the actual protection measured by the IASC method is less than the face value. Table 2 demonstrates that the IASCs derived by the linear programming method yield different rankings depending upon the external rate of return used and the year in which protection is required. The external rates of return considered are 5, 8, and 12 percent. The year in which protection is required is varied to include year 0 protection, year 10 protection, and year 20 protection. In each instance the horizon for analysis is 20 years. The case in which protection is required in year 20, shows the greatest variation in IASC as the external interest rate changes. When the external rate is 12 percent, the number of infeasible policies is 19. Several policies have exceedingly high IASCs.

The internal rates of return obtained by the LP method are given in Table 3. Note that this table considers external rates of 8, 10 and 12 percent. For the smaller interest rates of 4 and 5 percent, many of the policies are unbounded. This unboundedness is indicated by negative IASCs in Table 2. In other words, for these policies, the lower bound rate of return exceeds the external rate used for discounting. As demonstrated by Schleef[8], the LP marginal rate of return on policy equity is inversely related to the LP derived IASC. Consequently, the policy rankings obtained from Table 2 are identical to those obtained from Table 3. The intuition underlying the LP method is that as the external rate of return appropriate to the insured increases, the cost of insurance increases. A corollary explanation is that as the external rate of return increases, the insured incurs an increasing opportunity loss of return on policy equity. This opportunity loss on policy equity occurs because the insured foregoes the opportunity to invest at the external rate, but, in the process, obtains insurance protection. From the standpoint of the LP method, cost of insurance is measured either in terms of a level interest adjusted cost or in terms of rate of return loss on policy equity. A set of policies will have identical rankings for each of these two methods.

IV. Discussion and Summary of Results

The principal findings of this study are listed below.

* Unlike the traditional methods of interest adjusted surrender cost and

Linton's rate of return, the linear programming method of insurance cost

measurement does not require separation of pure insurance and policy

equity.

* In terms of the linear programming model, rate of return on policy equity

and interest adjusted surrender cost are equivalent policy measures.

* The use of small interest rates may lead to misleading IASCs.

Specifically, IASCs may be negative implying that the rate of return on

policy equity is identical to the external rate available to the insured.

* Policies differ widely in their ability to provide actual insurance

protection beyond the first year. Several policies examined in this analysis

were incapable of providing insurance protection required in year 20.

* Given that insurance protection is required in some year beyond the first

year, a policy may be incapable of providing actual insurance protection

if the external rate of return is sufficiently large.

The above findings raise several issues. First, is it necessary to periodically adjust the rate of return used for computing IASCs? The results of the above analysis indicate that some policies rank very high for small interest rates, but rank low for larger interest rates. This finding suggests that the interest rate should be adjusted to reflect prevailing economic conditions. For example the use of default free rates such as 20-year treasury bond rates may provide more meaningful policy rankings.

A second issue that arises from this study relates to which measure is preferred, the IASC or rate of return? The analysis of 68 policies suggests that if protection is required at the outset, then the IASC discriminates better among policies. The LP rate of return tends to be very close to the external rate in this case. Nevertheless, if the year of required protection is several years into the future, then either measure appears to be acceptable in terms of its discriminatory power. In this case, one may prefer the LP rate of return measure since it is readily compared with other financial instruments with similar risk characteristics.

A third issue raised by this analysis is that of requiring protection beyond the first year. Is it necessary or important to examine the ability of a policy to provide protection in later years? Since many individuals purchase insurance with the intent of keeping the policy in force for several years, it is appropriate that policy measures address protection in later years as well as the first. Many insurance professionals argue that frequently the insured's peak insurance needs occur 10 to 20 years from the time the policy is issued. Indeed, the traditional methods of measurement have always considered planning horizons several years into the future, e.g., 10 and 20 years. The LP method merely includes the additional dimension of year of required protection.

Fourth, given the importance of considering protection in later years, what is the appropriate way to measure this dimension of the policy? For a variety of reasons the industry has used 20 years as the planning horizon. It would, therefore, seem appropriate to measure the ability of a policy to provide protection twenty years hence. The upperbound rate of return on feasibility, derived from the LP method, provides such a measure. The larger this upperbound, the better the policy performs in terms of providing actual insurance protection several years hence. Table 4 indicates the upperbound rate of return for each of the 68 insurers analyzed in this study.

The linear programming method used in this study has been applied to whole life insurance. Nevertheless, the method may also be applied to other types of life insurance such as term insurance and interest sensitive products such as universal life. The application to term insurance is relatively straight forward. The main difference is that no terminal cash value is present, which implies that the lower bound rate of return cannot be computed. In order to apply the LP method to products such as universal life, it is necessary to agree on a common set of assumptions regarding the payment of premiums and the exercising of various policy options. The more general linear programming model developed in [7] may be applicable to the problem of selection and timing of options to exercise.

APPENDIX

In this appendix the linear programming model is presented. The fully developed model may be found in[8]. The following notation is used throughout the appendix. Linear programming decision variables:

Policy parameters and requirements of the insured:

[PLD.sub.t] = net premium rate in year t,(including adjustments for

[PROT.sub.t] = insurance protection required at the beginning of year t. For

The linear programming model is

Maximize [Mathematical Expression Omitted]

s.t. [PLD.sub.t]U + [W.sub.t] - [Z.sub.t] [is less than or equal to] [b.sub.t] for j = 1,2,...,n,

[Mathematical Expression Omitted],

U, [W.sub.t], [Z.sub.t] [is greater than or equal to] 0

If the above linear program is called the primal problem, then the corresponding dual linear program is given by

Minimize [Mathematical Expression Omitted]

s.t. [Mathematical Expression Omitted]

[Mathematical Expression Omitted],

[Beta.sub.t], [Mu.sub.t] [is greater than or equal to] 0. [Beta.sub.t] denotes the dual variable corresponding to the tth budget constraint and [Mu.sub.t] denotes the dual variable for tth insurance protection constraint. From an economic perspective, the [Beta.sub.t]'s represent marginal discount factors for each year whereas the [Mu.sub.t]'s represent the marginal discounted cost of increasing the death benefit requirement in each year. [Exhibit 1 Omitted]

(1)Projected dividends were interpolated for years not given.

REFERENCES [1]Belth, Joseph M., "The Rate of Return on the Savings Element in Cash Value Life Insurance," Journal of Risk and Insurance, Vol. XXXV (December 1968), pp. 569-81. [2]Best's Flitcraft Compend 1984, A. M. Best Company, Oldwick, New Jersey, 1984. [3]Hutchins, Robert C. and Quenneville, Charles E., "Rate of Return Versus Interest-Adjusted Cost," Journal of Risk and Insurance, (March 1975), pp. 69-79. [4]Ingraham,Jr., Harold G., "An Analysis of Two Cost Comparison Methods -- Interest Adjusted Cost vs. Linton Yield," Journal of the American Society of Chartered Life Underwriters, Vol. XXXIII, No. 4, (October 1979). [5]Linton, M. Albert, "Life Insurance as an Investment," in Davis W. Gregg (Ed.), Life and Health Insurance Handbook, 2nd Edition, Richard D. Irwin, Inc., Homewood, Illinois, 1964. [6]Report of the Joint Special Committee on Life Insurance Costs, report to American Life Convention, Institute of Life Insurance Association of America (New York: Institute of Life Insurance, May 4, 1970). [7]Schleef, Harold J., "Using Linear Programming for Planning Life Insurance Purchases," Decision Sciences, Vol. 11. No.3 (July 1980), pp. 522-34. [8]Schleef, Harold J., "The Joint Determination of Marginal Rate of Return and Interest Adjusted Cost for Whole Life Insurance," Management Science, Vol. 29, No. 5, (May 1983), pp. 610-21.

ABSTRACT

The traditional measures of Interest Adjusted Surrender Cost and Linton's rate of

return are computed for 68 whole life policies. Similar measures are obtained via linear

programming where cash flows are discounted using several different external interest

rates. With the linear programming method, year of insurance protection is varied to

include year 0, year 10 and year 20. If protection is required in year 20, several policies

are infeasible (incapable of generating any insurance protection). Upper limits for rates

of return are calculated. Infeasibility occurs for a given policy if the external rate of

return exceeds the upper limit.

Introduction

The purpose of this article is to use a linear programming (LP) method for measuring the cost of whole life insurance. The method is applied to policies offered in 1984 by 68 different insurers. For comparative purposes, the traditional methods of interest adjusted surrender cost (IASC) and Linton's rate of return are applied to the same set of policies. The insurers and policies were selected using criteria established in an earlier study by Hutchins and Quenneville [3]. Consequently, the results of this study of 1984 policies may be compared to a set of similar policies offered in 1972.

The LP method is similar to both IASC and Linton methods in several respects. Specifically, all three methods assume deterministic projected dividends and a fixed horizon of 20 years. The LP technique can be used to derive level interest adjusted costs similar to the IASC and internal rates of return on equity similar to the Linton rate of return. Nevertheless, both the IASC method and Linton's method rely on assumptions not required by the LP method. The LP method requires only an assumption of a rate of return that is relevant to the policyholder.

Conceptually, the IASC and Linton's method treat the whole life insurance policy as providing the two products of protection and savings. The IASC attempts to measure the level cost of protection while the Linton method attempts to measure the rate of return on the savings component. The IASC requires an interest rate for discounting whereas Linton's method requires various term insurance rates. As a consequence of these different required assumptions, rankings obtained by these two methods will not be perfectly correlated.

The LP method, by contrast, does not attempt to directly separate protection and savings. The method assumes that the insured individual requires a given level of protection. It is irrelevant to the insured how the insurer provides the protection. In other words, the insured is unconcerned with how the insurer divides the premium into loading charges, reserves, and so forth. The LP method has the additional flexibility of considering the point in time at which the insured requires protection. Neither the IASC nor Linton method explicitly address issues related to timing of insurance protection.

The flexibility of varying the year of required protection is the primary characteristic of the LP method that differentiates it from the traditional methods. The IASC, for example, implicitly assumes that coverage is required at the time the policy becomes effective. The IASC, however, does not recognize the reduction in the insured's wealth resulting from the premium payment. The LP method recognizes premium payments by increasing the policy face value by a corresponding amount. With the LP method, premiums are compounded and accumulated at the relevant interest rate to the year in which protection is required. This accumulation of premiums may be large if protection is required 15 or 20 years into the future. For large interest rates the accumulated premiums will, at some point, exceed the face value of the policy. The upper bound interest rate at which accumulated premiums equal exactly the face value of the policy will be presented below for the various policies studied.

I. Measures of Whole Life Insurance Cost

The interest adjusted surrender cost, recommended by the Joint Special Committee on Life Insurance [6], is computed by discounting each annual premium, per $1000 face value, by an appropriate interest rate. Premiums are reduced appropriately to reflect projected dividends, if applicable. The cash value at the horizon plus any terminal dividend is discounted and subtracted from the sum of discounted premiums. The resulting difference is divided by the annuity factor corresponding to the interest rate. The end result provides a measure of the level cost per $1000 of insurance.

A major problem with the IASC is the choice of an interest rate for discounting. Ideally the interest rate should reflect the return that the insured could realize on an investment similar to the equity inherent in the policy. The choice of interest rate should be related to the uncertainty of cash flows. The largest amount of uncertainty occurs from dividend payouts, since actual dividend payouts seldom match precisely the projected dividends. In other words, the uncertainty of dividend payouts introduces an element of risk into the policyholder's portfolio. Also, nominal interest rates vary considerably over time so that a rate used ten or 15 years ago may not be appropriate today. Yet little attempt has been made to adjust the rate used to compute IASC's reported in the trade periodicals. The only observable change is that, in recent years, IASC's have been computed using an interest rate of 5 percent rather than 4 percent. Clearly, even the risk free rate has exceeded 5 percent in recent years. As a result, the IASC's for many policies are negative. In addition to being intuitively unappealing, negative IASC's may be misleading. Because of the structure of premiums, dividends and ending cash values, the rankings of a given set of policies may differ when different interest rates are used.

Another problem associated with the IASC is that the amount of insurance protection is based upon the face value of the policy. Nevertheless, the effective amount of insurance protection inherent in a policy is actually less than the face value if the cash outlay of the premium is taken into consideration. By paying the premium, the insured has reduced his or her asset level and thereby his or her capacity to self insure. By ignoring this factor, the IASC of a policy can be decreased arbitrarily, for example, by increasing the premium and making appropriate increases in the horizon cash value.

A problem related to the effective amount of protection generated by a policy is the timing of protection required by the insured. Typically the horizon length for analysis and measurement purposes is either ten or 20 years. Setting the horizon at 20 years, however, does not guarantee that the policy under consideration is providing the face value of protection at the horizon. To obtain the precise amount of effective protection in a given year, the face value amount must be adjusted to reflect the accumulated premiums paid up to that year. It is possible, as demonstrated in the latter part of this article, that the premiums accumulated at interest exceed the face value of the policy. In essence, such policies provide no effective protection.

In contrast to the IASC, which measures the cost of protection given a rate of return, Linton's method measures the policy rate of return given the cost of protection. Although not commonly reported in trade publications, Linton's rate of return method is well known and documented in the literature. With this method, the actual insurance cost is separated from that part of the premium designated for policy equity. The computations are somewhat involved, but in the end an internal rate of return is obtained. The reader is referred to the original work of Linton [5] and an article by Belth [1] for details on computation of the rate of return. The derived rate of return provides a measure that can be compared to rates of return on other financial instruments. Nevertheless, some problems remain with the rate of return method. For example, the amount of pure insurance protection considered in an analysis varies as the trial rate of interest varies. The major problem with this method, however, is that it is necessary to assume term rates so that the insurance provided can be separated from protection provided by the policy equity. If high term rates are used, then the resulting rate of return will be higher, whereas if low term rates are used, then the resulting rate of return will be lower. How does one choose appropriate term rates for such an analysis? No one has satisfactorily answered this question. For example, Hutchins and Quenneville [3], who fully recognized this problem, used composite term rates. These composite term rates were computed by averaging the term rates for several insurers. As with the IASC, the rate of return method does not address the timing of protection.

The third method considered in this article, the linear programming method, imposes constraints for the insured and maximizes the discounted cash flows associated with a given policy. Two types of constraints are present. First, constraints are included to represent amounts that the insured is willing to budget for insurance. For purposes of developing comparative measures, these amounts are set to zero. Since the linear programming model includes variables for borrowing and lending, setting the budget amounts to zero forces the solution to include positive amounts for borrowing. The second set of constraints establishes limits on the amounts of protection required in each year to the horizon. For purposes of developing a cost measure, the analysis that follows includes only a single protection constraint corresponding to the year in which insurance is required. A single variable is used to represent the face value of the policy, purchased at the outset, necessary to satisfy all of the constraints. The complete mathematical linear programming formulation is included in the Appendix.

In effect, the LP method measures the ability of a given policy to provide a specified level of protection for a given number of years. That is, the LP method is concerned only with a specified level of protection rather than how the protection and savings components are separated within a given policy. The horizon length, however, remains the same as with the traditional methods (e.g., 20 years). With the LP method also, it is necessary to assume an interest rate for discounting. The interest rate used, however, is an external rate at which the insured can borrow and lend. Although different rates could be used for borrowing and lending, for simplicity it is assumed that these rates are identical. Since the budget amounts are equated to zero, only the borrowing rate is relevant. The negative of the optimal objective function value is the discounted cost of the policy over the 20 year horizon. Dividing this objective function value by the annuity factor corresponding to the external rate yields a level interest adjusted cost. An identical interest adjusted cost is obtained by observing the dual price corresponding to the single insurance constraint. This dual price can be interpreted as the discounted marginal cost of an additional dollar of insurance protection. Multiplying by 1000 and dividing by the 20 year annuity factor produces the level interest adjusted cost per $1000. The full derivation is provided in Schleef [8].

In addition to the insurance protection dual price, the LP method also uses the dual prices for the budget constraints. The dual prices corresponding to each of these constraints can be interpreted as a marginal discount factors. That is, the dual price for the budget constraint in year t denotes the present value of having one additional dollar available in year t. By summing these dual discount factors for each year to the horizon, an annuity factor is obtained. Finding the corresponding level interest rate for this annuity factor provides a marginal rate of return for the policy. Consequently, the LP model yields jointly an IASC and a rate of return, each of which is relative to the assumed external rate of return.

II. Characteristics of The LP Method

The precise inverse relationship between level cost of insurance and marginal rate of return is given in [8]. The level interest adjusted cost obtained with the LP method is similar to the traditional IASC. For example, if the protection requirement constraint for the first year is set equal to $1000, then the IASC derived by the LP method will differ from the traditional IASC by a constant factor. To demonstrate this point, if the first year premium is $18 per $1000 face value, then the LP model will call for $1018 of face value insurance. The traditional IASC will, in this case, equal 0.9823 times the LP IASC, i.e., 0.9823 = 1000/1018. The linear programming method, however, allows the flexibility of requiring a given level of protection in years other than the first. By setting the budget constraints to zero, the effective amount of protection generated is measured rather than the face value amount of the policy. As the year in which insurance is required occurs farther into the future, the corresponding level cost of insurance increases. This happens because the model requires that the initial policy is sufficiently large to cover the premiums up to the year in which the protection is required. In addition, the premiums are compounded at the (external) interest rate used. This effect will be demonstrated later in the paper for a selected policy.

The marginal rate of return obtained from the LP method is relative to the external rate used for borrowing and at which the cash flows are discounted. This marginal rate of return does not measure directly the rate of return of the equity component of the policy, but rather provides a marginal rate of return on the policyholder's wealth relative to the policy. It can be shown that the marginal rate of return derived from the LP method will never exceed the external rate of return used for borrowing and discounting. The difference between the marginal rate of return and the external rate represents the loss, in terms of rate of return, that the insured incurs as a result of holding the insurance policy.

Each policy has two limiting rates of return related to the LP method that warrant discussion. Both of these limits apply to the external rate of return used for discounting. The lower limit rate of return corresponds to the external rate of return at which the LP solution becomes unbounded. That is, for external rates of return below this limit, the objective function increases without bound. The interest rate at which this occurs satisfies the following equation where [PLD.sub.t] denotes the premium less dividend for year t, [CV.sub.n] denotes the cash value plus any dividends at the horizon, and i denotes the interest rate.

[Mathematical Expression Omitted] The limiting rate of return, i, is the internal rate of return that occurs if the cash flows associated with the policy are evaluated without any consideration for the insurance inherent in the policy. As the external rate of return approaches this lower limit, the marginal rate of return derived from the dual LP discount factors also approaches this lower limit. The lower limit for rate of return is the internal rate of return realized if only the equity component of the policy is considered independent of the face value insurance provided in the policy.

The upper limit on the external rate of return represents that rate of return at which the linear programming solution is infeasible. Infeasibility implies that the policy under consideration is incapable of providing any actual insurance protection. Such infeasibility occurs if the premiums, compounded at the external rate of return, exceed the face value of the policy. For example, if insurance is required for the 20th year, infeasibility occurs if the compounded sum of the premiums exceeds the face value of the policy. Mathematically infeasibility occurs if

[Mathematical Expression Omitted] where j denotes the latest year protection is required.

Since the first year premium never exceeds $1000 per thousand dollars of face value, infeasibility never occurs when the insurance requirement occurs for the first year. As the insurance requirement occurs farther into the future, the external rate of return at which infeasibility occurs decreases.

III. Application of The Linear Programming Model

In this section, the LP method is used to compute comparative measures for 68 whole life policies. The results of the LP analysis are compared to results obtained by using the traditional measures of IASC and Linton rate of return.

The 81 insurers considered by Hutchins and Quenneville [3] were used as a basis for selecting policies for analysis. The data used in the present study were collected from Best's Flitcraft Compend 1984 [2].(1) The criteria used for selection were also similar to those used by Hutchins and Quenneville (henceforth referred to as H & Q). The present study considers policies of $25,000 for males age 35 compared to policies with a face value of $10,000 considered by H & Q. Of the 81 insurers considered by H & Q in 1972, 68 reported policy data in Best's Flitcraft Compend 1984 [2] meeting the various criteria. For several, minor differences from those policies considered by H & Q may exist. For example, 36 of the 68 policies analyzed below are for nonsmokers. In order to compute Linton's rate of return, term rates are required. Data from ten non-participating five- year renewable term policies were collected. Seven of the term policies were standard policies while the remaining three policies were for nonsmokers. The rates reported for males age 35, 40, 45, and 50 were averaged separately to obtain the following term rates. These rates are used for the age given and the four subsequent renewal years.

To provide a basis for comparison with the H & Q study, Table 1 lists the rate of return and IASCs for each insurer. The IASCs are computed using interest rates of 4, 5, 8, 10, and 12 percent. H & Q used an interest rate of 5 percent--the most commonly used rate in the industry--in their study. Note that IASCs reported in Table 1 are computed using projected dividends for policies offered in 1984 whereas, H & Q also examined the actual historical performance of policies.

One of the main results of the H & Q study was the large negative correlation between IASC and Linton rate of return. Ranking the 68 policies on these two measures yields a rank order correlation coefficient of -.938 compared to -.979 for the H & Q study. Substantial differences in the levels of each of these two measures exist between the two studies. For example, the term rates used in the present study are approximately one third less than the corresponding rates used by H & Q. Nevertheless, the mean Linton rate of return is 7.82 percent with a standard deviation of 2.02 compared to a mean of 4.75 percent and standard deviation of 0.73 reported by H & Q for policies issued in 1972. Using an interest rate of 5 percent yields a mean IASC of $1.44 per thousand with a standard deviation of 2.44 for the 68 insurers in this study. This compares with a mean of $6.01 and standard deviation of 1.07 in the H & Q study.

As reported above, the 1984 Linton rates of return are higher and the IASCs are lower than in the H & Q study. Moreover, Table 1 demonstrates that different rankings of insurers occur as the interest rate used for discounting is changed. One of the conceptual problems with the IASC method is choosing an appropriate rate for discounting. Given the default free rates prevailing in 1984, such as T-bond rates, a rate of 5 percent is too low. If the IASC in some sense measures cost of insurance, then negative IASCs suggest negative costs of insurance. To the extent that insurers and insureds compete in the same investment market, it is unlikely that life insurance will have a negative cost. Consequently, a more reasonable interest rate for computing IASCs should, at a minimum, yield positive IASCs. For example, Table 1 indicates that interest rates of 8 percent and above yield positive IASCs for each insurer.

The various cost measurement methods discussed in this article will be demonstrated by an example. Exhibit 1 provides such an example. The policy data pertain to a $25,000 policy issued by Connecticut Mutual Life. The IASCs computed by the traditional method indicate that a policy's ranking depends upon the interest rate used for discounting. For example, the given policy has the smallest IASC if a 4 percent rate is used, but the policy ranks 35th out of 68 if a rate of 12 percent is used. The policy is ranked 25th (tie) for the Linton return. The linear programming measures yield rankings similar to the traditional method, given that small interest rates are used. This is not surprising, especially for the case of requiring protection at the outset (year 0). In the year 0 protection case the LP IASC differs from the traditional IASC by a constant. That is,

LP IASC = Trad. IASC/(1-0.02155) where the cost per $1000 of face value (protection) is $21.55. In effect, the LP measure requires additional insurance to cover the initial premium. In this example, it is necessary to purchase $1022.02 of face value in order to generate $1000 of protection, i.e.,

$1022.02 = 1000/(1-0.02155) For those cases in which protection is required in later years, such as years 10 or 20, the LP IASC and the traditional IASC also differ by a constant. But in these cases the constant accounts for insuring against the compounded premiums paid up to and including the year of protection. Schleef[8] provides the formula to compute the LP IASC for the general case.

Even though the traditional IASCs and the LP IASCs differ by a constant for each policy, the policy rankings for these two methods are not identical. For example, with an interest rate of 12 percent, the traditional method results in a ranking of 35 for the Connecticut Mutual policy. With the LP method, using an external interest rate of 12 percent yields a ranking of 36 for year 0 protection, but a ranking of 40 if protection is required in year 10. Moreover, if protection is required in year 20, no feasible solution exists. This means that the sum of the first 20 premiums per $1000 protection, adjusted for projected dividends and compounded at 12 percent, exceeds $1000 by the beginning of year 20. Feasibility occurs only if the interest rate is less than the upperbound of 11.74 percent (see Table 4). The upperbound limit of 11.74 percent ranks this policy 51st in terms of the upperbound rate, given protection is required in year 20. In contrast, the lower bound interest rate of 6.14 percent is relatively high with a ranking of 4. This means that the cash flows resulting from premiums, projected dividends, and terminal cash value will yield 6.14 percent over a 20 year period. To summarize, the Connecticut Mutual policy ranks first in terms of the traditional IASC, given the low interest rates of 4 and 5 percent, but is incapable of providing any actual insurance protection if the interest rate is 12 percent and protection is required in year 20.

The LP method of analysis requires that an external interest rate be specified. This external rate represents the insured's opportunity cost of foregoing investment so that insurance may be purchased. The second factor included in the linear programming analysis is the year in which insurance protection is required. With the Linton and IASC methods the face value protection occurs at the outset, but there is no guarantee as to the level of insurance protection in subsequent years. If the amount of the premium must also be covered, then the actual protection measured by the IASC method is less than the face value. Table 2 demonstrates that the IASCs derived by the linear programming method yield different rankings depending upon the external rate of return used and the year in which protection is required. The external rates of return considered are 5, 8, and 12 percent. The year in which protection is required is varied to include year 0 protection, year 10 protection, and year 20 protection. In each instance the horizon for analysis is 20 years. The case in which protection is required in year 20, shows the greatest variation in IASC as the external interest rate changes. When the external rate is 12 percent, the number of infeasible policies is 19. Several policies have exceedingly high IASCs.

The internal rates of return obtained by the LP method are given in Table 3. Note that this table considers external rates of 8, 10 and 12 percent. For the smaller interest rates of 4 and 5 percent, many of the policies are unbounded. This unboundedness is indicated by negative IASCs in Table 2. In other words, for these policies, the lower bound rate of return exceeds the external rate used for discounting. As demonstrated by Schleef[8], the LP marginal rate of return on policy equity is inversely related to the LP derived IASC. Consequently, the policy rankings obtained from Table 2 are identical to those obtained from Table 3. The intuition underlying the LP method is that as the external rate of return appropriate to the insured increases, the cost of insurance increases. A corollary explanation is that as the external rate of return increases, the insured incurs an increasing opportunity loss of return on policy equity. This opportunity loss on policy equity occurs because the insured foregoes the opportunity to invest at the external rate, but, in the process, obtains insurance protection. From the standpoint of the LP method, cost of insurance is measured either in terms of a level interest adjusted cost or in terms of rate of return loss on policy equity. A set of policies will have identical rankings for each of these two methods.

IV. Discussion and Summary of Results

The principal findings of this study are listed below.

* Unlike the traditional methods of interest adjusted surrender cost and

Linton's rate of return, the linear programming method of insurance cost

measurement does not require separation of pure insurance and policy

equity.

* In terms of the linear programming model, rate of return on policy equity

and interest adjusted surrender cost are equivalent policy measures.

* The use of small interest rates may lead to misleading IASCs.

Specifically, IASCs may be negative implying that the rate of return on

policy equity is identical to the external rate available to the insured.

* Policies differ widely in their ability to provide actual insurance

protection beyond the first year. Several policies examined in this analysis

were incapable of providing insurance protection required in year 20.

* Given that insurance protection is required in some year beyond the first

year, a policy may be incapable of providing actual insurance protection

if the external rate of return is sufficiently large.

The above findings raise several issues. First, is it necessary to periodically adjust the rate of return used for computing IASCs? The results of the above analysis indicate that some policies rank very high for small interest rates, but rank low for larger interest rates. This finding suggests that the interest rate should be adjusted to reflect prevailing economic conditions. For example the use of default free rates such as 20-year treasury bond rates may provide more meaningful policy rankings.

A second issue that arises from this study relates to which measure is preferred, the IASC or rate of return? The analysis of 68 policies suggests that if protection is required at the outset, then the IASC discriminates better among policies. The LP rate of return tends to be very close to the external rate in this case. Nevertheless, if the year of required protection is several years into the future, then either measure appears to be acceptable in terms of its discriminatory power. In this case, one may prefer the LP rate of return measure since it is readily compared with other financial instruments with similar risk characteristics.

A third issue raised by this analysis is that of requiring protection beyond the first year. Is it necessary or important to examine the ability of a policy to provide protection in later years? Since many individuals purchase insurance with the intent of keeping the policy in force for several years, it is appropriate that policy measures address protection in later years as well as the first. Many insurance professionals argue that frequently the insured's peak insurance needs occur 10 to 20 years from the time the policy is issued. Indeed, the traditional methods of measurement have always considered planning horizons several years into the future, e.g., 10 and 20 years. The LP method merely includes the additional dimension of year of required protection.

Fourth, given the importance of considering protection in later years, what is the appropriate way to measure this dimension of the policy? For a variety of reasons the industry has used 20 years as the planning horizon. It would, therefore, seem appropriate to measure the ability of a policy to provide protection twenty years hence. The upperbound rate of return on feasibility, derived from the LP method, provides such a measure. The larger this upperbound, the better the policy performs in terms of providing actual insurance protection several years hence. Table 4 indicates the upperbound rate of return for each of the 68 insurers analyzed in this study.

The linear programming method used in this study has been applied to whole life insurance. Nevertheless, the method may also be applied to other types of life insurance such as term insurance and interest sensitive products such as universal life. The application to term insurance is relatively straight forward. The main difference is that no terminal cash value is present, which implies that the lower bound rate of return cannot be computed. In order to apply the LP method to products such as universal life, it is necessary to agree on a common set of assumptions regarding the payment of premiums and the exercising of various policy options. The more general linear programming model developed in [7] may be applicable to the problem of selection and timing of options to exercise.

APPENDIX

In this appendix the linear programming model is presented. The fully developed model may be found in[8]. The following notation is used throughout the appendix. Linear programming decision variables:

U = face value of insurance purchased at the outset. [W.sub.t] = amount lent externally by the insured at the beginning of year t. If the budget amounts (defined below) are equated to zero, the LP solution will never call for lending. [Z.sub.t] = amount borrowed externally by the insured at the beginning of year t.

Policy parameters and requirements of the insured:

[PLD.sub.t] = net premium rate in year t,(including adjustments for

projected dividends.) [CV.sub.t] = cash-value rate (policy equity per dollar of face value insurance) at the end of year t. [b.sub.t] = amount budgeted by the insured at the beginning of year t. For the purpose of developing policy measures, the budget amounts are set to zero.

[PROT.sub.t] = insurance protection required at the beginning of year t. For

the purpose of developing policy measures, a single insurance protection amount (corresponding to the latest year for which protection is required) is set equal to 1000. The constraints f or all other years are dropped. i = external rate of return used for discounting. [v.sub.t] = present value of $1 received at the beginning of year t, i.e., [v.sub.t + 1] = [(1 + i).sup.-t] for t = 0,1,...,n n = number of years to the end of the planning period.

The linear programming model is

Maximize [Mathematical Expression Omitted]

s.t. [PLD.sub.t]U + [W.sub.t] - [Z.sub.t] [is less than or equal to] [b.sub.t] for j = 1,2,...,n,

[Mathematical Expression Omitted],

U, [W.sub.t], [Z.sub.t] [is greater than or equal to] 0

If the above linear program is called the primal problem, then the corresponding dual linear program is given by

Minimize [Mathematical Expression Omitted]

s.t. [Mathematical Expression Omitted]

[Mathematical Expression Omitted],

[Beta.sub.t], [Mu.sub.t] [is greater than or equal to] 0. [Beta.sub.t] denotes the dual variable corresponding to the tth budget constraint and [Mu.sub.t] denotes the dual variable for tth insurance protection constraint. From an economic perspective, the [Beta.sub.t]'s represent marginal discount factors for each year whereas the [Mu.sub.t]'s represent the marginal discounted cost of increasing the death benefit requirement in each year. [Exhibit 1 Omitted]

(1)Projected dividends were interpolated for years not given.

REFERENCES [1]Belth, Joseph M., "The Rate of Return on the Savings Element in Cash Value Life Insurance," Journal of Risk and Insurance, Vol. XXXV (December 1968), pp. 569-81. [2]Best's Flitcraft Compend 1984, A. M. Best Company, Oldwick, New Jersey, 1984. [3]Hutchins, Robert C. and Quenneville, Charles E., "Rate of Return Versus Interest-Adjusted Cost," Journal of Risk and Insurance, (March 1975), pp. 69-79. [4]Ingraham,Jr., Harold G., "An Analysis of Two Cost Comparison Methods -- Interest Adjusted Cost vs. Linton Yield," Journal of the American Society of Chartered Life Underwriters, Vol. XXXIII, No. 4, (October 1979). [5]Linton, M. Albert, "Life Insurance as an Investment," in Davis W. Gregg (Ed.), Life and Health Insurance Handbook, 2nd Edition, Richard D. Irwin, Inc., Homewood, Illinois, 1964. [6]Report of the Joint Special Committee on Life Insurance Costs, report to American Life Convention, Institute of Life Insurance Association of America (New York: Institute of Life Insurance, May 4, 1970). [7]Schleef, Harold J., "Using Linear Programming for Planning Life Insurance Purchases," Decision Sciences, Vol. 11. No.3 (July 1980), pp. 522-34. [8]Schleef, Harold J., "The Joint Determination of Marginal Rate of Return and Interest Adjusted Cost for Whole Life Insurance," Management Science, Vol. 29, No. 5, (May 1983), pp. 610-21.

Printer friendly Cite/link Email Feedback | |

Author: | Schleef, Harold J. |
---|---|

Publication: | Journal of Risk and Insurance |

Date: | Mar 1, 1989 |

Words: | 5695 |

Previous Article: | Perceived risk: the link to plan selection and future utilization. |

Next Article: | Risk aversion and the value of information. |

Topics: |