Variable annuities and variable life insurance

JSS 20 (2) (1972) 69-112

VARIABLE ANNUITIES AND VARIABLE LIFE INSURANCE IN THE UNITED STATES OF AMERICA by

S. HAMILTON LECKIE INTRODUCTION THE purpose of this paper is to outline the development and current status of variable annuities and variable life insurance in the United States of America. The author was fortunate in being granted a Fellowship by the Winston Churchill Memorial Trust which enabled him to travel extensively in North America for two months in the summer of 1971. It is pointed out that this paper is the result of a large number of impressions formed by the author and has no claim to be a comprehensive treatise on the subjects. However, it is hoped that the paper will be of real interest to actuaries and others in the United Kingdom. Part I of the paper deals with variable annuities, Part II with variable life insurance, and Part III with the special problem of providing minimum death benefit guarantees and maturity value guarantees for these variable products. Variable annuities are much more established in the United States than in this country, but variable life insurance is just in the process of being developed. Mention will be made of the broader issues as well as of actuarial matters. Background information

There are many differences in the conditions under which life insurance companies in the U.S. operate compared with the U.K. U.S. life offices are subject to much greater regulation and the regulations are applied by each of the 50 states, plus the District of Columbia. Before a life office can write business in any state, it must be licensed and have its insurance policies approved by that state's insurance department. The insurance laws differ in certain respects from state to state and there are many small offices which 69

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are licensed in only a few states. Agents, that is the salesmen, must become licensed by examination before selling insurance in any state. The state insurance departments lay down minimum cash surrender value and valuation bases and also in some cases, the terms of the policy contract. The taxation authority in the U.S. is the Internal Revenue Service (I.R.S.) which operates on a federal basis. However, some of the states also charge a tax of up to 3% on the premium income derived from business written in that state. Life offices may operate their own mutual funds (U.S. equivalent of unit trusts) and any company writing variable annuities or mutual fund business is also regulated by the Securities and Exchange Commission (S.E.C.). This federal body regulates anything deemed to be a 'security'. Conventional fixed dollar life insurance and annuities are not regulated as securities, but variable annuities are so regulated and the position of variable life insurance is not yet settled. The purpose of the S.E.C. is to protect the small investor by, for example, demanding minimum standards of disclosure and by fixing maximum amounts of commission. The S.E.C. is not always easy to contend with and inconsistencies arise in the S.E.C.'s dealings with different companies and from year to year. However, the S.E.C. does seem to be a necessary part of financial life in the U.S. In general, a U.S. company which is considering introducing a new type of policy must expend much time and effort on the regulatory implications. Only once these constraints have been satisfied can the company start to think of the purely actuarial matters such as setting premium rates. Another major difference is that American actuaries are more marketing minded than their British counterparts and the proprietary companies seem to be more profit-conscious. PART I. THE VARIABLE ANNUITY Development and regulation

The variable annuity may be defined as an annuity under which the annuity benefit varies with the investment performance of a separate account and the annuitant bears the investment risk. The contract provides for the annuitant to pay premiums during an accumulation period and these contributions are invested in a separate account. The annuity period commences at the end of the accumulation period and the annuitant thereafter receives periodic

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payments, the cash amount of which depends on the market value of the separate account at the time of payment. An immediate variable annuity is the special case with no accumulation period. The first variable annuity in the United States was the College Retirement Equities Fund (C.R.E.F.) system. This was established in 1952 as a companion organization to the Teachers Insurance and Annuity Association, a non-profit-making body providing insurance only to teachers. Concern about retirement security had led to a comprehensive study of techniques of combating the inflationary erosion of pension dollars. The system, which has proved very successful, has been fully described in British actuarial literature by MacLean (1) and by Blunt and Lane (2). The principle behind the establishment of C.R.E.F. was that a variable annuity would firstly provide protection against inflation by maximizing the amount of funds available at retirement and secondly, provide an increasing benefit which would not be eroded by the fall in the value of the dollar. Many economists and others have demonstrated that historically the value of a portfolio of ordinary shares with net income reinvested has, over the long term, increased by more than the cost of living although there is no correlation between share prices and the cost of living in the short term. Clearly, however, the yield on a fixed dollar annuity could rise to a point where future increases in the variable annuity payments are fully discounted. In the mid-1950s a few small insurance companies were formed to sell variable annuities without having much impact on the industry. However, the S.E.C. became interested in variable annuities being offered to the public because of the similarities of the contract to a mutual fund during the accumulation period. After a series of court cases it was established that the variable annuity is a security and therefore comes under the jurisdiction of the S.E.C. This decision has had far-reaching consequences for the insurance companies as it meant that the variable annuity insurer has to contend with four overlapping spheres of regulatory controls, viz. the S.E.C, the state insurance departments, the state security commissions and the Internal Revenue Service. However, the regulatory position has gradually become a little clearer although the variable annuity is still subject to dual regulation as a security and as an insurance policy. The whole development of the variable annuity has been well documented by Campbell (3).

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The requirements of the S.E.C. are extremely comprehensive and may be considered in 3 parts, viz. regulation of the variable annuity contract, of the separate account, and of selling practices. The purpose of the regulations is to ensure that complete disclosure is provided on the nature of the variable annuity and to prevent misrepresentation in the sale of the product. Before a sale is made the agent must deliver a prospectus to the purchaser. Prospectuses can be extremely lengthy and contain comprehensive information on the contract including a description of the expense loadings and the charges on the separate account, the options available, the rights of the policyholder and financial information on the separate account and on the company. Prospectuses must be approved word for word by the S.E.C. and this can be a very long process. The S.E.C. controls the maximum amount of commission payable and the level of the expense loadings. Every agent is required to pass an examination on the securities business, as are specific head office personnel such as field supervisors and investment managers. The agent is also strictly controlled on any illustrations used to indicate the possible benefits of the policy. Illustrations can only be based on past performance of the separate account, which places new companies at a disadvantage. The state insurance departments require an insurance company to be specially licensed to sell variable annuity business and the requirements for licensing vary from state to state. Agents and associated head office personnel may be required to take an additional examination on variable annuities. The end result of all these requirements is delay and expense for the company since the registration with the federal and state authorities may take from 6 months to 2 years. The delays in the development of the variable annuity were not all of external origin. Within the insurance industry itself a major philosophical battle waged for a decade as to the desirability of variable annuity business. Some companies felt that the policyholder expects security and certainty from life insurance contracts, that it is the business of insurance companies to insure the investment risks, and that there would be dangers in the manner of the selling of these contracts. Furthermore it was argued that the life insurance industry should concentrate on using its influence to counter inflationary pressures in the national economy. However, other companies insisted that the variable annuity is in fact a sound

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insurance policy which is needed by the public, that inflation is beyond the control or influence of business, and that the variable annuity would help the purchaser avoid the consequences of inflation through participation in the growth of the economy. Criticism of the variable annuity has now disappeared and most of the large- and medium-sized companies have entered the variable annuity market but the attitudes of different companies still vary considerably. Some companies are very enthusiastic about the variable annuity as part of their portfolio of products, but other companies only entered the market as a defensive measure for reasons of competition. The attitude of any particular company is reflected in its sales force. In some companies 100% of the agents are licensed to sell variable annuities whereas in other companies the proportion is as low as 25%. Also the attractiveness of this business to the agent is affected if the amount of commission paid is less than under the office's conventional annuities, either because of the S.E.C. or by design. Types of variable annuity

There are four main types of variable annuity in the United States: 1. Group variable annuities, used for pension business. 2. Variable annuities available only to self-employed persons (called H.R. 10 or Keogh plans after the legislation introducing the favourable tax treatment of these plans). 3. 'Tax sheltered' variable annuities for school teachers. 4. Individual non-qualified variable annuities which are available to the public. In general, the first 3 categories of contract are 'qualified' by the I.R.S. and therefore the company pays no tax on the separate account backing this type of business. However, non-qualified separate accounts are subject to tax on long-term capital gains. Hence at least 2 separate accounts are desirable if not essential for a company writing both qualified and non-qualified business. Many pension plans in the U.S. are of the money purchase type and there are no regulatory restrictions on the relationship between the amount of the pension and final salary. Consequently most group variable annuities are simply a collection of individual policies with premiums being received in bulk. H.R. 10 and tax sheltered annuities form very attractive business to insurers since there is B

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tax relief available to the policyholder and the average policy size is large. Some companies specialize in this highly competitive market. The individual variable annuity market is not yet fully developed. Here the insurance companies run into direct competition with mutual funds and other savings media and at the same time the agent's commission is much less than for life insurance. On the other hand individual variable annuity contracts are not at a disadvantage as they are in the U.K. compared with life insurance in that neither life insurance nor annuity premiums attract tax relief. Many companies have been somewhat disappointed in their efforts to sell individual variable annuities. A fixed part of an individual non-qualified annuity is regarded as return of capital and this 'excludable amount' is tax-free in the hands of the recipient. As in the U.K., the method of taxation results in the net benefits under a variable annuity fluctuating less than the gross benefits. If the benefit under a variable annuity is less than the excludable amount in any year, the excess can be respread over future years to increase the tax-free element. Each of these classes of variable annuity is, of course, available in fixed dollar form and many companies stress the idea of a balanced annuity in which part of each premium, not necessarily constant, is applied to a fixed dollar policy. The balanced annuity provides some guarantee of dollar amount and protects the benefits in a period of deflation. Also the volatility of the annuity benefits is lessened but at the cost of a reduction in the equity participation. At the vesting date there is often an option to change the composition of the balanced annuity. It is also usual to guarantee that at least the premiums received to date will be returned in the event of death during the deferred period. Interest rates are currently high in the United States and so the initial benefit on an immediate variable annuity compares unfavourably with the return on a fixed dollar annuity. As in the U.K., few immediate variable annuities have been sold. However, the policyholder is not given the full benefit of the high interest rates in the premium basis for a deferred fixed dollar annuity and so the variable annuity becomes more attractive the longer the deferred period. Premium basis Interest. No interest assumption is, of course, made for the accumulation period. During the annuity period it is not necessary

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to make an interest assumption either, but it is universal practice to do so. Theoretically the accumulated moneys could merely be placed in the separate account and the expectation of life used to determine the initial benefit. The initial payment would then be much lower than under a fixed dollar annuity although the investment performance of the separate account with regard to both income and capital gains would be fully reflected in the subsequent annuity unit values and hence in the annuity benefits. To overcome the problem of a low initial payout, an interest rate called the Assumed Investment Return (A.I.R.) is introduced. The initial benefit is obtained by dividing the accumulated moneys by the appropriate annuity factor based on the A.I.R. The A.I.R. anticipates a certain rate of return in the separate account and the actual investment performance of the separate account is reflected in the annuity unit value only to the extent that the earned rate of interest exceeds the A.I.R. The A.I.R. thus determines the initial benefit and also the rate at which the benefits increase for a given change in the value of the underlying assets. If the A.I.R. is too high then there is the possibility that payments may not increase as rapidly as desired, or may even diminish over the long term. Conversely, if the A.I.R. is too low then the initial benefit will compare unfavourably with the benefit under a fixed dollar annuity. Should the A.I.R. coincide with the dividend yield on the separate account, then the capital gains or losses in the separate account will be directly reflected in the annuity unit value. The A.I.R. used in the U.S. ranges from 3½% to 5% although some group contracts may use up to 6%. The state insurance departments regulate the maximum A.I.R. which may be used and in some states this is only 3½%. Mortality. Mortality assumptions for the variable annuity tend to follow those for fixed dollar annuities. In the deferred period fairly conservative mortality guarantees are given using projection mortality tables. The guarantees are usually arranged so that a deduction of one year in age is made for every complete 15 or 20 years by which the year of birth is later than 1900. In the annuity period little or no margin is included in the mortality assumptions, unless the group plans are of the experience refund type or the individual variable annuities are participating; the variable annuities offered by some companies are fully or partly participating with respect to mortality and expenses. Expenses. The expenses of the variable annuity are considerably

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higher than for fixed dollar annuities because of extra costs, (a) arising from the handling of the separate account and in the daily evaluation of unit prices, (b) incurred because of the amount of regulation and the need to issue a prospectus, and (c) owing to clerical staff being unfamiliar with the variable contract. Notwithstanding these facts, the expense loadings are often taken the same as for fixed dollar annuities, thus achieving consistency at the expense of equity. It should be remembered that the S.E.C. will not approve a contract if the loadings are too high. Sales charges, which include commission, are limited by the S.E.C. to the equivalent of a level annual charge of 9%, but many offices pay less commission than the maximum. American actuaries are not so concerned with inflation of costs as in Britain and many believe that increased efficiency and automation will offset the effect of higher salaries and costs; some U.S. companies have actually reduced the per policy renewal expenses in the last 10 years. The mortality and expense guarantees during the deferred period are usually paid for by an annual charge on the assets. The risk to the company under these guarantees depends on the amount of the assets at vesting and so the risk can be matched if the reserve for the guarantees is allowed to accumulate within the separate amount. Lapses. Premium bases in the U.S. usually include assumptions as to lapses. The assumptions are fixed with regard to each office's own experience and account may be taken of the loss (or profit) to the company in the event of withdrawal in each policy year. No lapse assumptions are, of course, made for annuities after vesting. Profit. In general, American offices do not expect to make any profits from mortality and also expenses are running higher than charged for. However, a percentage charge is always made on the assets for investment management expenses and usually for the guarantees, and a 1% annual charge on large funds can yield a significant amount. Few companies, if any, have yet reached the point where expenses are being paid for by the direct loading plus the asset charge. The profit to the company is expected to come from the annual charge, and as the assets build up, the charge should help offset any inflationary increases in renewal expenses. The U.S. approach to premium rates is highly sophisticated, at least in the large companies. Once the basic assumptions have been made specimen rates can be calculated (this may be regarded as the microscopic approach, i.e. individual lives are considered in the

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usual manner). The next step is to consider a model office over a 20- or 30-year future period. The projections incorporate detailed sales forecasts and the emerging cash flow can be calculated by allowing for lapses, amount put to reserve, mortality, etc. The cash flow is then discounted. Stock market performances can also be simulated and some models allow for the lapse rates to vary according to stock market fluctuations. High lapses combined with unfavourable investment experience in the separate account could result in the life insurance company suffering a substantial loss. On the other hand, with a favourable persistency experience and good investment results the life office would appear to be able to achieve a very attractive return. Mutual companies expect a return of about 10% per annum to justify investing current surplus in the development costs of the variable annuity and with proprietary companies the rate may be nearer 15% per annum. Methods of measuring profitability other than by calculating the return on capital employed are also used. As a result of the projections, the premium rates may be adjusted, that is the microscopic rates may be adjusted as a result of macroscopic considerations. The key to the profitability of the variable annuity lies in the asset charge. The asset charge in fact replaces the excess interest earnings which the insurance companies are accustomed to receiving on fixed-dollar policies. A charge of between ½% and 1½% per annum seems small to the average policyholder who tends to pay more attention to the direct deduction from his premiums. In fact, this small percentage can build up to provide a very large income for the insurer without the insurer taking any of the investment risk. However, the investment performance of the separate account is, of course, reduced by the amount of the charge. Reserves

During the accumulation period the assets are automatically equal to the liabilities other than for the guarantees given. The minimum death benefit guarantee will be considered in Part III. Theoretically, the charges made for the annuity rate guarantees should be accumulated right up to the vesting date but often no specific reserves are set up in which case the charges would go directly into surplus. In the annuity period the reserve in units for each policy is equal to the number of units payable per annum multiplied by the appropriate annuity factor at the then attained age. Some of the

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states set a maximum valuation rate of interest of 3½%, but because of the mechanics of the variable annuity no valuation strain arises from the interest element irrespective of the value of the A.I.R. This may be demonstrated as follows. Consider a variable annuity payable annually in arrear to an annuitant currently aged x, with the next payment due in a year's time. Let K = dollar amount at the time of valuation of the annual payment under the annuity. Let i = A.I.R., and suppose the valuation rate of interest is 3½%. If interest at the valuation rate is earned each year in the future, the amount in dollars of the annuity payment one year hence would be

and in two years' time would be

and so on. Then the reserve on the valuation basis is

= Kax at rate of interest i. If the annuity had just been purchased, then K would have been determined by dividing the consideration by ax at the A.I.R. (ignoring expenses). Hence no strain arises on purchase if the variable annuity is valued at the A.I.R. as though it were a fixed dollar annuity for the same annual amount as the current dollar value of the benefit under the variable annuity. The separate account

The separate accounts underlying variable annuities in the U.S.

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seem to be almost entirely invested in common stocks (i.e. U.S. ordinary shares) with the balance in government securities or cash. Each separate account has a board of directors who are elected by the policyholders and any change in the investment policy of the separate account must be approved by the policyholders. Some companies feel that a separate account invested in property would be very attractive but the supervisory authorities would very likely impose severe controls on the level of liquidity and on valuation procedures. The same investment portfolio can form the assets for both qualified and non-qualified contracts although theoretically there might be differences in the investment philosophy since a nonqualified account is taxed on long-term capital gains. However, in some companies at present capital gains are more than offset by operating losses so that no tax is paid on the separate account. There also would appear to be a case for using different accounts for the accumulation and annuity periods, since in the latter a major consideration should be to try to minimize fluctuations in the annuity payments whereas in the former the only objective is to maximize the investment return. Some companies use averaging of the unit price during the annuity period so as to smooth out dayto-day fluctuations. Current situation

The variable annuity can be criticized on the grounds that it does nothing to guarantee that the proceeds will not be eroded by inflation. However, no insurer appears to have developed a feasible means of pricing and marketing an annuity with benefits unconditionally linked to the cost of living. It is, of course, possible to design an annuity which increases at the rate of inflation but subject to a maximum rate of, say, 3% per annum. In general the cost of such increasing annuities makes them appear unattractive compared with fixed dollar annuities. It can be argued that the current high yields on immediate fixed dollar annuities do in fact partly reflect inflation. However, one fact that is quite certain is that the real value of fixed dollar benefits will decrease every year there is inflation; in other words a fixed dollar annuity is effectively a decreasing annuity in real terms. The variable annuity is now fairly well established in the United States and the rate of new developments has slowed down. Some

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companies have not met their initial expectations, but partly this has been the result of recent uncertainties in the stock market; in fact in May of 1971 redemptions of U.S. mutual funds exceeded sales for the first time. Another difficulty is that some of the older agents have been reluctant to sell variable annuities after a lifetime of promoting the 'security' of fixed dollar benefits. One principle that has been clearly demonstrated is that for any office to succeed in selling variable annuities it is essential first to sell the idea to the sales force. PART II. VARIABLE LIFE INSURANCE

Variable life insurance may be defined as life insurance with benefits which vary according to the investment performance of a separate account. Premiums may be fixed or variable and there is a wide variety of ways of linking the benefits to the separate account. Before discussing the development of true variable life insurance in the United States, a description will be given of four existing types of policy. Existing types of policy The "cost of living" rider. This is a rider to a whole life or term insurance policy under which additional insurance may be purchased each year without evidence of health for an amount sufficient to restore the real value of the policy. This benefit is not funded, so that the premium under the policy increases each time additional insurance is bought and the risk to the company is limited to an element of mortality anti-selection. There are also policies under which the dividends (i.e. the annual bonuses declared on participating business, usually paid in cash) are applied as paid-up additions or as term insurance to the extent of the inflation over the year, before the balance, if any, is paid in cash. Package products. Some U.S. life offices have their own mutual funds or allow their agents to sell other mutual funds. It is possible to combine term insurance with systematic investment in the mutual fund. This is similar in effect to the typical United Kingdom unitlinked policy where a small percentage of the premium pays for life cover and expenses and the balance is used to purchase units of a unit trust or internal account. However, because of regulatory

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restrictions, two contracts must be issued in the U.S. although one cheque can be used to pay for both contracts. Either the term insurance or the investment portion can be discontinued without affecting the other. In some offices the investment medium could also be the accumulation account of a deferred variable annuity. However, the offices do not actively encourage this type of contract. One reason is that such package products produce substantially less commission for the agent than does traditional life insurance. Also the investment portion is regulated by the S.E.C. and the boom days of mutual fund sales may be over. Although unit-linked policies are proving very successful in the U.K., the Americans are seeking to develop true variable life insurance in which the entire benefits vary with the separate account investment performance. Equity funding. There is available an interesting contract called an Equity Funding Plan under which a lump sum or systematic investment is made in any one of several mutual funds. Insurance premiums are then loaned to the policyholder using the mutual fund shares as collateral security. There are strict regulatory requirements as to the margin between the value of the shares and the accumulated loan. As well as life insurance, it is possible for other types of insurance to be paid for in this way. Clearly to benefit from the Equity Funding Plan the average rate of return on the mutual fund shares must exceed the rate of interest being paid on the loan. Life insurance linked to the consumer price index. The best example of this is a whole life policy with premiums payable throughout life as described by Bragg and Stonecipher in a paper to the Society of Actuaries (4). The sum assured under the policy increases each year by the same amount as the increase in the consumer price index, but premiums remain level. Thus a policy will maintain its real value while the policyholder is not required to pay increased premiums, so that the inflation risk is borne by the life office. The only limit on the increases is that the maximum benefit is twice the initial sum assured, but if the cost of living should fall in any year, the sum assured is not reduced. In determining premium rates, an assumption as to the rate of inflation must be made. In this case the assumption was 3½% for the first five years and 2½% thereafter, so that the maximum sum assured is assumed to be reached after 27 years. This select approach to estimating rates of inflation was made after studying historical changes in the cost of living and considering the current situation.

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The company, by assuming the inflation risk, stands to make a loss if inflation runs higher than the estimated rates. Higher rates of inflation would bring forward the time when the maximum sum assured under any policy is reached and consequently the company would pay out greater than expected benefits on death within 27 years of the commencement of the policy. On the other hand, if inflation runs at lower levels than predicted, the company stands to profit. The risk to the company of higher than expected inflation is not significant at the younger ages; the premiums under the policy are more sensitive to variations in the number of lapses than to variations with the same estimated probability of occurrence in the rate of inflation. Also, it seems feasible that rapid inflation may encourage persistency and there is the possibility that the higher inflation will be accompanied by higher interest earnings. In the 3 years since the policy was introduced, the consumer price index has in fact increased by more than 3½% per annum. The provision of cash surrender values was a problem since non-forfeiture values must be guaranteed in the U.S. With an index-linked policy, surrender values can either be fixed at the outset or expressed per thousand of current sum assured. There was also a problem in determining when the Standard Non-Forfeiture Law was satisfied. Bragg and Stonecipher decided on cash values and reserves fixed per policy. It was proved that the maximum policy value at any point in time occurs if the consumer price index has not increased at all up to that time but then instantaneously doubles. The surrender values were based on these policy values and as the amounts were greater than produced under the premium basis, the cost of surrenders had to be taken into account in the premium basis. Another problem was that the paid-up sum assured of the policy exceeds the initial sum assured at advanced durations, so that the paid-up amount eventually becomes the minimum death benefit irrespective of the actual course of the consumer price index. Some of the advantages of the consumer price index-linked policy are: (a) It gives the cost of living protection directly, rather than relying on investment performance to approximate the effect of inflation. (b) No separate account is necessary. (c) This product is clearly life insurance, so that the S.E.C. is not involved.

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id) Normal commission patterns can be followed. (e) The policy could be issued under existing insurance laws in most states. (f) The company is assuming the investment risk. On the other hand, the policy has certain drawbacks: (a) There is a ceiling on the sum assured. (b) The policyholder foregoes any possibility of superior investment performance from the separate account. (c) The premium is on average about 165% of the premium for a fixed benefit whole life policy with the same initial sum assured. (d) The principle behind the policy has not always proved easy to explain to prospective policyholders. One powerful criticism of cost-of-living policies is that the life insurance cover required by any individual policyholder will change over the period of one year by an amount which bears little or no relation to the change in the consumer price index. However, this does not absolve the insurer from his responsibility to provide benefits which remain valuable to the policyholder in an inflationary era. Unfortunately, few companies have shown much interest in index-linked insurance. Development of true variable life insurance

In November 1969, Fraser, Miller and Sternhell presented a paper to the Society of Actuaries entitled 'Analysis of Basic Actuarial Theory for Fixed Premium Variable Benefit Life Insurance' (5). In the words of the authors, the paper 'presents an analysis of the basic actuarial theory for life insurance policies which have (i) fixed premiums, (ii) the entire reserve held in a separate account, the assets of which would be invested primarily in common stocks, and (iii) benefits adjusted to reflect the investment performance of the separate account in such a manner that the policy owners would bear the entire investment risk and the life insurance company would not share any part of the investment risk'. The method described in the paper relates the benefits to the separate account in such a way that the reserve per dollar of actual sum assured at the end of each policy year is exactly the same as for a corresponding fixed benefit policy. The sum assured at the end of any policy year is obtained by multiplying the sum assured

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at the preceding policy anniversary by a Y factor, representing the adjustment to reflect the fact that a fixed premium is payable, and a Z factor representing the adjustment to reflect the actual investment return on the separate account compared with the Assumed Investment Return (A.I.R.). The actuarial theory can be developed as follows by considering a fixed benefit whole life policy with unit sum assured. The equation connecting successive terminal reserves is (1)

= reserve after rryears years of a whole life policy issued at agex. Px = net annual premium for a whole life policy issued at agex. i = A.I.R. qx+t-i = rate of mortality at age x + t—l. Consider a fixed premium variable benefit whole life policy with unit initial sum assured and with the same annual premium as the corresponding fixed benefit policy. Suppose that the sum assured of this policy is adjusted as described above. Then we have

where

(2)

where and

F r = face amount at the end of the r'th policy year i't = actual net investment return on the separate account over the t'th year.

Equation (2) can be expressed as Applying equation (1) to the right-hand side of this expression we have

Solving for Ft, we obtain

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If we let and then Using this relationship to connect successive sums assured gives the desired property that the reserve per dollar of actual face amount at the end of each policy year for the fixed premium variable benefit policy is exactly the same as for a corresponding fixed benefit policy, i.e. reserve after t years = Ft(,Vx). This result is verified using a prospective valuation method in the Appendix. The theory can be easily applied to other forms of insurance. Furthermore, the formula can be extended to policies under which the premiums vary in any specified manner, or where the death benefit under the corresponding fixed benefit policy varies in any given manner. Note that for a paid-up policy, Y, is unity, and so Ft = F t - 1 Z t , i.e. the change in the sum assured depends only on the Z factor. There is an alternative approach using unit principles. Let u0 = unit value of separate account at the commencement of the policy, UT = unit value of separate account at the end of the t'th policy year. If the unit values are adjusted to reflect the actual investment earnings of the separate account over the A.I.R. then

Now let Xo = initial number of units of sum assured, X, = number of units of sum assured at the end of the t'th policy year. Consider a fixed premium variable benefit whole life policy with an initial sum assured of 1 and with an annual premium of Px. Then F o =X 0 u 0 = 1 and Ft=X,M,.

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Substituting in equation (3) we obtain

Since this reduces to (4)

Equation (4) defines the recursion process required to determine the change in the number of units of sum assured from the end of the (t— l)th policy year to the end of the t'th policy year. In order to keep the number of units of sum assured constant each year, the premium payable would have to vary according to the changes in the unit value. There is complete symmetry between the two approaches. Since F,_ x = xX t_t-1 1Ut-1 equation (4) can be written

i.e. Also i.e. The significance of the Y and Z factors in the basic equation Ft=Ft-1YtZt can now be expressed as (a) the role of the Y factor is to adjust the number of units of sum assured (and hence the sum assured itself) at the beginning of the t'th policy year to reflect the fact that a fixed premium of Px is payable at that time, (b) the role of the Z factor is to adjust the sum assured so as to reflect the change in the unit value over the t'th policy year.

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It should be noted that there is no need to introduce numbers of units or unit values and actual sums assured can be determined solely from the Y and Z factors. As may be expected, the Fraser, Miller and Sternhell paper attracted a great deal of attention in the U.S. and this is reflected in the discussion to the paper. It transpired that various actuaries had been thinking along similar lines and it seems appropriate to summarize the reasons why the idea of variable life insurance should become of more than just theoretical interest in the U.S. at this particular time. The need for variable life insurance 1. Inflation has been part of life in the United States since World War II. The rate of inflation has averaged about 2½% per annum over the last 25 years although the increase in the cost of living in 1970 was almost 6%. Consequently, there has been the problem that 'good' dollars have been paid for life insurance but the benefits will only be received in 'devalued' dollars and the public is generally aware of the erosion in the value of fixed dollar benefits. Some American economists feel that inflation will markedly decrease as a result of the present Administration's measures, but others are not so confident and predict that inflation over the next 25 years will continue at levels at least as high as in the past. 2. In 1945 the assets of the life insurance companies represented over 50% of all institutional savings but this percentage has fallen steadily to the current level of approximately 18%. Very little endowment business is now written and the insurance companies have lost ground to other savings media such as mutual funds, noninsured pension funds, banks and savings and loan associations (U.S. equivalent of building societies). The life companies have concentrated on providing pure life protection without providing a complementary investment service. However, the industry would like to recover some of the lost ground and this has been witnessed in the emergence of variable annuities and in the setting up of mutual funds by some life companies. 3. Participating policies are very common in the U.S. but the bonus loadings are smaller than in the U.K. at about 10% to 15% of the premiums. Also life office investment in ordinary shares is limited to 10% of the general assets or only 5% in some states. Consequently life insurance dividends have not reflected the increased

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earnings of equities. However legislation has gradually been passed by the various states to exempt separate accounts from the investment restrictions on the general assets. 4. The method of declaring dividends in the U.S. hinges on the dividend formula by which is calculated the amount of dividend for each policy according to its contribution to surplus in each of the three elements of mortality, expense and interest. In recent years the interest earned by the companies on new money has been much higher than the return on the existing assets, but many life offices use the same earned interest rate in the dividend formula for all policies regardless of year of commencement and this is inequitable to the newer policies. One of the advantages of using a separate account approach is that new policyholders can have the full benefit of the prevailing high investment returns. Alternative designs of variable life insurance

Many alternative approaches to variable life insurance were suggested in the discussion to the Fraser, Miller and Sternhell paper. Basically the excess earnings of the separate account can be used to increase benefits in any desired manner. For example, paidup benefits or term insurance could be bought with the excess earnings and the additional benefits may be either variable or fixed. The Fraser, Miller and Sternhell design is in fact the particular case where the excess earnings are used to increase the sum assured in such a manner that the policy reserve per unit of sum assured is the same as for a corresponding fixed benefit policy. For any given time series of investment results the pattern of benefits produced by the various designs differs widely. Six of the more important designs are: 1. Dutch (this type of policy first appeared in the Netherlands around 1953) This is a fully variable policy under which the premiums, sum assured and reserves are expressed in units rather than dollars and directly reflect the relationship between the actual investment performance of the separate account and the A.I.R. 2. New York Life (Nylic) (as described in the Fraser, Miller and Sternhell paper) This is the fixed premium counterpart of the Dutch design under

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which the reserve per unit of sum assured, as in the case of the Dutch policy, is the same as for a corresponding fixed dollar policy. 3. Fairbanks This is a policy under which a portion of the premium is used to purchase fixed dollar one year term insurance equal to the excess of the initial sum assured over the amount of reduced paid-up insurance purchased by the reserve on a corresponding fixed dollar policy and the balance of the premium is used to purchase variable paid-up insurance. 4. Walker This is the counterpart of the Fairbanks design where the oneyear term insurance is variable rather than fixed. 5. Cooper This is a 'buy term and invest the difference' policy under which a portion of the premium is used to purchase fixed dollar one-year term insurance equal to the excess of the initial sum assured over the reserve on a corresponding fixed dollar policy and the balance of the premium is put on deposit in the separate account. 6. Booth This is the counterpart of the Cooper design where the one-year term insurance is variable rather than fixed. A ctuarial formulae Let

where i't = actual net investment return on the separate account over the t'th year and i = A.I.R. Sums assured 1. Dutch 2. Nylic 3. Fairbanks c

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4. Walker 5. Cooper 6. Booth Reserves

1. Dutch 2. Nylic 3. Fairbanks 4. Walker 5. Cooper 6. Booth There is, of course, no 'correct' design. Each alternative has different characteristics and places different relative emphasis on increasing death benefits compared with increasing cash values. Also the extra benefits can be made fixed or they can remain variable. Some designs are much more volatile than others. The Nylic design, for example, will give a much greater increase in the early years in death benefits for a given favourable investment return in the separate account than the Walker design, but the converse is true if the separate account investment return is inferior to the A.I.R. Regulation

Variable life insurance has to date not been issued in the U.S. and the principal reason for this is that the regulatory requirements have not been determined. Traditional life insurance policies are subject to regulation and variable life insurance will be regulated also. The major issues are 'by whom' and 'to what extent' and these problems are still being resolved. During 1970 the American Life Convention and the Life Insurance Association of America, whose member offices transact over 99% of U.S. life insurance business, set up a joint committee to formulate a united approach to variable life insurance regulation. In October of 1970 this committee made an informal submission to the Securities and Exchange Commission on behalf of the life insurance industry.

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The purpose of the submission was to demonstrate to the S.E.C. that variable life insurance is not a security and therefore that the S.E.C. should in no way be involved in the regulation of variable life insurance. This united approach by the industry to variable life insurance regulation contrasts markedly with the situation which occurred with variable annuities. In that precedent a few companies started selling variable annuities and then were taken to court by the S.E.C. with the ultimate result that the variable annuity now suffers dual regulation. Specifically the submission proposes that the S.E.C. should not exercise jurisdiction over variable life insurance policies which have four basic characteristics designed to ensure that the insurance purpose and the insurance function of such variable life policies will be predominant. The four characteristics are: 1. 'The contract must provide lifetime insurance coverage'. Thus endowment and term business are excluded. 2. 'The contract must be issued for an initial stated amount of death benefit and must guarantee payment of a death benefit of at least equal to such amount.' Such a minimum death benefit guarantee is clearly necessary for the policyholders' protection. 3. 'The amount payable upon the death of the insured in any year must be no less than a minimum multiple of the gross premium payable in that year by a person who meets standard underwriting requirements.' By excluding policies having very high premiums or a short payment term, this requirement seeks to limit the investment element of variable life policies. 4. 'The entire contract must be a life insurance contract subject to regulation under the state insurance laws, including all required approvals by state insurance commissioners.' The implication here is that state insurance regulation is sufficient to protect the consumer. The S.E.C. responded to the submission by asking for more information on variable life insurance and this was supplied by the A.L.C.L.I.A.A. Committee in February, 1971. At the end of July, 1971 the S.E.C. asked the industry to make a formal submission which will be followed by a hearing. It now seems likely that no conclusion on the regulatory position will be reached before the middle of 1972. Full regulation by the S.E.C. would entail the issuance of pros-

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pectuses, limitation of charges, registration of agents, regulation of sales practices and regulation of the separate accounts. The insurance companies would of course prefer the S.E.C. not to regulate but this may be an unrealistic hope. However, if the S.E.C. does decide to regulate then the industry would press very strongly for exemption from the 1940 Investment Company Act. This is the act which limits the amount of commission which can be paid on the sale of securities. Clearly variable life insurance will not be attractive to the agency force if the maximum commission payable is limited by the S.E.C. to below that payable on conventional life policies. Some but not all of the states have already modified their insurance laws so as to permit the issue of variable life. Many of the states which currently permit variable life require disclosure to the buyer of specified information about the policy and the issuing company, but not so fully as in a variable annuity prospectus. The question of whether the state securities departments will become involved depends on the outcome of the negotiations with the S.E.C. There are also unresolved questions about the taxation of variable life both at the company and the individual level. Specifically, the treatment of the investment gains of the separate account in the taxation of the insurance company and the applicability of current taxation rules on the policy proceeds to the policyholder or beneficiary are complex issues. Current issues

Although most of the industry is very much in favour of the introduction of variable life insurance, there is still a substantial body of opinion which is opposed to variable life, or at least has some reservations. Many of the arguments which follow were previously voiced at the time when the variable annuity was the subject of debate. 1. Probably the main objection is to the assumption that variable life insurance with assets invested primarily in ordinary shares will provide the best protection against inflation. In the final analysis share prices may depend on corporate profits and there is no guarantee that inflation will be beneficial to profits. It is argued that the life insurance industry should devote its resources towards developing policies which are guaranteed to

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maintain their real value. The ultimate in policy design may be a policy under which the death benefits vary with the cost of living while the cash value is tied to equity performance, but this would be almost impossible to achieve in the U.S. because of the regulatory position. 2. There are substantial fears that variable life might be mis-sold. It is felt that there should be additional controls on the methods of selling variable life which can be used by the agent, especially as the public will not understand how benefits are related to the performance of the separate account. Clearly the policyholder must appreciate the volatility of the benefits and that in contrast to conventional policies, surrender values are not guaranteed as to dollar amount. It is perhaps unfortunate that proposals for self-regulation are not in the American tradition, but the prevailing U.S. attitude is that self-imposed constraints are an infringement on their freedom of action and that only the law need be obeyed. 3. With conventional life insurance, the principal source of profits and surplus to the life office is the excess interest earnings. However, with variable life, superior investment results are passed directly to the policyholder and it may be asked why the proprietary companies should offer variable life if they cannot make as much profit as with conventional business. Clearly the answer to this lies in the pricing of the contracts, especially in the recurring charge on the assets, but it seems possible that the introduction of variable life insurance will not benefit the smaller companies, some of which will not generate sufficient business to recoup their development costs. 4. There are fears about the effect on the economy of variable life insurance. At present only estimates can be made of the volume of business which will be written, but it is conceivable that variable life could form 50% of all new whole life business within a few years. Large-scale investment in ordinary shares by the insurance companies might result in a shortage of stock which would push prices artificially high and disturb the nation's capital market. At the same time, the life insurance companies would reduce their holdings of fixed interest securities and cut back on their supply of mortgage money with the result that interest rates could rise considerably. In reply to this reasoning, it is pointed out that variable

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annuities have had no noticeable effect on the economy, although the predictions on the impact of variable life are altogether on a much larger scale. Almost certainly, variable life will have some effect on the economy, but the magnitude of the effect can only be guessed at. In the event it may be that the investment patterns of the life insurance companies will change so gradually that any effect on investment patterns will not be detrimental. In general, it may be said that most of the arguments against variable life insurance are made out of concern for the industry as a whole. However, each individual office will make its decision on whether or not to introduce variable life on the basis of its own interests. Current situation The current position is that at least three of the large U.S. companies are well advanced in their plans to market variable life insurance. Almost every other company is following developments avidly without proceeding too far before the outcome of the negotiations with the S.E.C. An excellent review of the development of variable life insurance is given in Variable Life Insurance: Current Issues and Developments (6). The three designs of variable life which will certainly be sold are the Nylic design, the Dutch design and the Walker design. One company has had its policy documents for the Nylic and Dutch designs approved by several of the state insurance departments. It is interesting to compare the three main designs. Most companies are assuming that fixed premiums are a prerequisite of variable life insurance but this may not be true. The Dutch design is the only one with variable premiums and this has several interesting consequences. If the separate account performs well, then the policyholder will be required to pay a larger premium and if the account performs badly then a smaller premium will be asked for. This may have a beneficial effect on lapses, but the design contravenes the idea of price cost averaging. From the company viewpoint, favourable investment performance will result in a larger expense loading being received, and vice versa. The variable premium design will also affect the cost of the minimum death benefit guarantee. The Nylic design is more volatile than the Walker design and so good investment performance will be more obviously reflected in

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the sum assured. However, since the sum assured under the Nylic design depends on both the Y and Z factors, it is possible for the Z factor to be greater than unity but the sum assured to decrease over the period of one year. It may be difficult to explain to the policyholder why his sum assured has decreased while the separate account earnings were greater than the A.I.R. The Walker design is akin to a participating policy in which dividends are declared in paid-up form, the main difference being that negative paid-up additions are inevitable in some years. The pricing of variable life insurance cannot be finalized until it is known whether the S.E.C. plans to regulate, particularly with regard to commission. However, it is anticipated that variable life premiums will be marginally dearer than for comparable fixed benefit policies. The A.I.R. is likely to be 3% and variable life policies offered by the mutual companies will probably be participating in the mortality and expense elements. Mortality assumptions will be the same as for fixed benefit policies. However, one difference with variable life is that favourable investment performance increases the amount at risk. Consequently, it may be necessary to use more severe underwriting standards. Reinsurance presents a particular problem since the amounts at risk cannot be pre-determined and have the potential of increasing greatly. The expenses of variable life will be higher than normal because of the development costs and the increased running costs of the separate account. To operate variable life successfully, it will be almost essential to be highly computerized, which may not be easy for small companies. Again, the profitability to the company of variable life will depend mainly on the asset charge. The development of variable life insurance is currently in a most interesting phase. No further steps can be taken to issue the first policy until the negotiations with the S.E.C. are resolved, but most offices are taking advantage of the hiatus to examine every aspect of variable life, even if no commitment has been made to proceed with the development of it. The marketing impact of variable life can only be guessed at. Every shade of opinion can be found, from high hopes that this will be the greatest innovation for decades, to predictions of dismal failure. One of the principal unknowns is the extent to which sales of variable life will represent a real increase in new business rather than just a switch from conventional life insurance. Since the initial sum assured

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will be guaranteed on death and the premiums will be almost the same as for conventional life, a policyholder would seem to have everything to gain and nothing to lose from a variable life policy. Although surrender values will not be guaranteed in a variable life contract, there is the possibility of substantially improving on the guaranteed surrender values in a comparable fixed dollar policy. On the other hand, it will be difficult to explain to a policyholder exactly how the benefits vary. Part of the appeal of the typical U.K. unit-linked policy is the very simplicity of the contract, but this will not be so with variable life insurance. Clearly the agents will have to be carefully trained on the correct means of selling variable life policies. PART III. MINIMUM DEATH BENEFIT GUARANTEES AND MATURITY VALUE GUARANTEES

The basic principle behind variable annuities and variable life insurance is that the policyholder assumes the investment risk. Nevertheless, it seems desirable that the insurer should provide some guarantee as to the benefits which will be paid. Guarantees on death or maturity would seem to be of real value to the assured so long as they can be provided at reasonable cost. At the same time the insurance company, by undertaking a financial risk, will receive a premium for doing so which gives the insurer additional opportunity for profit providing the correct premiums are charged. It is better for the insurance company to increase premium income rather than to reduce liabilities. However, there are many unresolved questions about these guarantees, particularly about the maturity value guarantee. Current situation

Most variable annuities in the U.S. provide, sometimes optionally, that on death during the deferred period a return will be made of the premiums paid to date or the value of the units whichever is higher. The guarantees may be attached to both single premium and systematic policies, and usually the benefit ceases at age 65, or on earlier vesting. No office appears to be offering any form of guarantee of the value on vesting or earlier surrender with variable annuities, although a guaranteed cash option at vesting of all premiums paid to date would seem to be an attractive selling point.

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All U.S. variable life insurance contracts will incorporate a minimum return on death of the initial sum assured; this is one of the conditions laid down in the submission to the S.E.C. No asset value guarantees in variable life policies are at present contemplated, but again there is scope for including a guaranteed surrender value at, say, age 65. It is possible that if variable life is a success, the trend will be to introduce variable endowment policies, perhaps with a maturity value guarantee. One insurance company does provide a maturity guarantee in conjunction with certain mutual funds, but the response by the public has been weak, mainly because the charge for the guarantee, at 6% of all money invested, seems very high. In Canada the situation with respect to variable insurance is more akin to the U.K. than to the U.S. Equity-linked endowment and deferred annuity policies which provide guarantees on death and maturity are commonly sold. In 1970 the Ontario Securities Commission indicated that certain types of equity-linked products would come under the supervision of that body if they did not contain benefits at death or maturity that would at least equal 75% of the gross premiums payable. The securities commissions in some of the other provinces are moving towards similar requirements and most companies have chosen to include 75% or 100% guarantees in their products. The problems

With both types of guarantee the twin problems are the calculation of premiums and the determination of suitable reserves. Traditionally, actuaries have always considered mean values in their assumptions, but with these types of guarantee, it is necessary to consider the whole range of possible outcomes of investment performance and attach probabilities to each value. The questions of premiums and reserves may be considered from three different viewpoints. Theory It is quite safe to say that not nearly enough theoretical work has been done on the nature of these guarantees, which provide some highly intriguing problems. The techniques which are appropriate such as mathematical statistics, risk theory and computer simulation, are not generally familiar to most actuaries.

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Regulation The only concern of the regulatory authorities is to ensure solvency for any company offering these guarantees. Office practice The viewpoint of the offices writing these guarantees is that premiums should be adequate but competitive. Reserves also should be adequate, but at the same time the company does not want to over-reserve or have violent fluctuations in the amount of its reserves. It should be borne in mind that the reserve for these guarantees will form only a very small part of the liability of most offices. Premiums for minimum death benefit guarantee 1. The simplest approach to calculating premiums for the minimum death benefit guarantee under a variable annuity is to assume that the separate account unit values will grow at a constant rate of, say, 5% per annum. The value of the units attaching to any policy is thus known for each policy year and the excess of the gross premiums over this value is the estimated liability to the company on death. The cost of the benefit can be calculated for any age at entry and single or annual premiums obtained. The premium is usually expressed as a percentage of the office premium. This method takes no account of fluctuations in the unit value, but is certainly very convenient. However, the method is useless for variable life insurance, since an assumed rate of growth higher than the A.I.R. implies that no payments would be made under the guarantee. 2. Perhaps the commonest method by which the premiums for the guarantee have been calculated for variable annuities is using a historical approach. A representative ordinary share index over a long period of time is chosen and adjusted as appropriate to reflect the inclusion of net reinvested income, capital gains tax and the charge against the assets. The net allocation to the separate account under any policy is then assumed to be invested at these prices and similarly the value of the policy in every subsequent year can be found. For each year of entry, a series can be formed of the amount which the company would have paid out under the guarantee. Hence net premiums can be calculated for each year of commencement. The office premium can be taken as the average of the historical net premiums plus a contingency loading. Alternatively a premium

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can be chosen on the basis that it would have been adequate in, say, 90% of all commencement years in the past. 3. A more sophisticated method is to apply computer simulation to obtain series of stock market prices. Simulation may be defined as a process which gives possible outcomes of an event by applying given data in a random manner. Using as data the changes which have occurred from year to year or from month to month in an adjusted ordinary share index, the value of an investment after a period of time can be simulated any number of times and a frequency distribution built up. A series can then be obtained of the amount payable under the guarantee and the benefit costed as before. This method is appealing from a theoretical viewpoint and may be regarded as producing a premium free from the irregularities of actual history. Most U.S. companies are intending to use simulation in pricing the minimum death benefit guarantee for variable life insurance. An alternative approach to simulation has recently been put forward by Bailey (7). The approach is one of enumeration rather than simulation and does not involve the generation of random numbers. Bailey uses a computer program named 'Dice' which calculates the complete frequency distribution of the expected sums of the faces which turn up when any finite number of dice are rolled; each die can have any number of faces, each die can have any amount on any of the faces and each die can be biased in any way. By using the historical changes of a stock market index, a complete frequency distribution of the value of an investment of 1, either made once only or systematically, can be obtained using this statistical technique. 4. It is possible to calculate the premiums for a minimum death benefit guarantee by representing stock market fluctuations as a mathematical function and applying risk theory. Kahn (8) has recently suggested that stock market price changes may be approximately represented by a log-normal curve. In his paper Kahn compares the single premiums for the minimum death benefit guarantee under a paid-up whole life policy obtained by simulation and by this analytic method. The advantages of the analytic approach are that it provides more precise measurements of the fluctuations of the stock market and that it is less expensive than simulation. It is salutary to examine the nature of the assumption made with the last three methods. The performance of a general stock index

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over perhaps the past 50 years is being used to estimate the performance of a particular separate account over the next 50 years. A comparison may be made with the early insurance companies which had to use population mortality statistics when setting premium rates. Fortunately at that time mortality was generally improving and the offices could exercise selection over the lives to be insured. It would seem to be highly improbable that the factors which determine investment returns will produce, over the next 50 years, results bearing the same characteristics as in the past. However, the first stage is certainly to examine past history, and this can only be done by looking at a suitable general index. The next step should be to adjust the distribution of investment results so as to fit in with the expected characteristics and the investment philosophy of any particular separate account. For example a property fund is generally regarded as being much less volatile than an equity fund, although this may not be true if the fund is concentrated in comparatively few individual properties. However, it will be very many years before anything other than one of the standard stock indices can be considered suitable. Premiums for maturity value guarantee 1. To estimate the value of the net allocations to a separate account in n years' time, an average growth rate could be chosen and the net allocations accumulated at this rate. However, this approach is totally inadequate since any reasonable growth rate will result in an estimated maturity value higher than the guarantee, implying that there is no liability. What is required is the distribution of possible maturity values, not just the mean. 2. As with the minimum death benefit guarantee, an historical approach is possible. Using a general stock index, adjusted as necessary, it is straightforward to calculate the maturity value for a policy of any term which commenced in any past year. Hence the amount, if any, which the company would have had to have paid out can be calculated. The average of these amounts can be divided by xn to give the average net annual premium. Since the nature of these guarantees is not yet fully understood, it is prudent to include a large contingency loading in the office premium. It is sometimes said that a maturity value guarantee is purely a financial risk, whereas a minimum death benefit guarantee combines financial and mortality risks. This is not really true, since the

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maturity value guarantee is only paid out on survival to maturity. However, the minimum death benefit guarantee is much more sensitive to a change in mortality rates than is the maturity value guarantee. Thus premiums for a minimum death benefit guarantee vary much more by age than the premiums for a maturity value guarantee. The cost of the latter guarantee is, however, markedly reduced by lapses. 3. The premiums can be calculated by developing the statistical distribution of the maturity value of a policy. This is the approach used by Turner in his prize-winning paper 'Asset Value Guarantees under Equity-Based Products' (9). Turner's paper considers three areas: (a) analysis of the probability density function of rates of return on ordinary shares, (b) presentation of a general simulation model for evaluating the net risk premium for the guarantee, and (c) determination of the net risk premium. The results in the paper are so presented that the sensitivity of the net risk premium to a change in the term of the policy, in the mean rate of return on the separate account and in other factors, can be easily appreciated. 4. An alternative approach was put forward by Di Paolo in 'An Application of Simulated Stock Market Trends to Investigate a Ruin Problem' (10). An investment risk premium for the guarantee is chosen, e.g. 1% or 2% of the gross premium and the 'risk fund' remaining after the maturity of the last contract of a block of business is calculated. The risk fund in this case is the accumulation of the premiums received for the maturity value guarantee less the amounts paid out under the guarantee, together with mortality profits less losses from the minimum death benefit guarantee. Using simulation, a distribution function of the risk fund can be established and hence the probability of ruin calculated. Di Paolo regards an investment risk premium as adequate if the probability of the risk fund being in a state of ruin after the last contract matures is less than about 10%. Reserves for minimum death benefit guarantee 1. The most obvious method of reserving for this benefit is to accumulate premiums received less benefits paid out. This method

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is simple, but hardly adequate as the reserve could become negative and no account is taken of future benefits. 2. A one-year term insurance reserve could be held for an amount at risk, if positive, of the guaranteed benefit less, say, 75% of the current value of the units. This method probably gives adequate reserves overall. It is extremely unlikely that the separate account unit value will drop sufficiently over the period of one year to give an average fall of 25%, but on the other hand, no account is taken of benefits in future years. This omission is more important for singlepremium contracts than for systematic policies, where the implicit assumption is that the future premiums for the guarantee will be at least sufficient for the future benefits. The method also suffers from the disadvantages that an initial strain is inevitable and that the reserve can fluctuate greatly from year to year. However, it should be borne in mind that the amount of the reserves for the guarantee may be relatively insignificant. 3. In 1970 Hickman presented a paper to the Society of Actuaries entitled 'A Reserve Basis for Guaranteed Benefits under Variable Annuity Contracts' (11) Hickman's paper reflects the belief that most of the 51 jurisdictions in the United States would be unwilling at this time to base reserves on the result of simulations, but would instead require an approach more similar to traditional methods. His approach is deterministic rather than probabilistic, not because it is theoretically better, but because it may be more acceptable to the state insurance departments. Reserves for both minimum death benefit and maturity value guarantees are considered by introducing change factors, which are analogous to the interest factor in normal life insurance reserves. Applying a change factor fn to the current unit value gives the expected unit value after n years on a conservative basis. In the determination of what specific change factors should be assumed for reserve purposes, w-year change factors were calculated from historical data for values of n between 1 and 40. The change factors chosen should be smooth to avoid discontinuities in reserve requirements, and the factors should be conservative so as to provide adequate reserves. The factors were chosen as

and these are the net change factors after allowing for charges of 1 % per annum against the assets. The gross change factors

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are compared with the lowest historical result, the 10th percentile, the 20th percentile and the median values. Fn is consistently more conservative than the 20th percentile and at durations 1-2 and over 20 is more conservative than even the 10th percentile. Hickman recognizes the disadvantages of both retrospective and prospective approaches; the former fails in that no account is taken of the current investment situation or of the future benefits whereas the latter produces reserves which depend on the unit value at the date of valuation and which can therefore fluctuate considerably. An average reserve is suggested which moves one-fifth of the way from the accumulation of the previous reserve towards the new prospective reserve, thus smoothing out the fluctuations to some extent. However, even with this modification the prospective method would historically have resulted in fluctuations in the reserve greater than the fluctuations in the claims arising under the guarantee. Reserves for maturity value guarantee This is a more difficult problem than the reserves for the minimum death benefit guarantee since the entire liability to the office is unknown until the very end of the contract. With a non-profit endowment policy, the insurer knows that he must pay out, say, £1,000 at the end of the term. The two end-points of the reserve curve are known and only the precise shape of the curve remains to be determined by the actuary. However, with a maturity value guarantee, only the starting point is fixed and the ultimate liability may be zero or it may be several hundred pounds. A colourful analogy may be drawn with a guided missile aiming at a target. With a conventional endowment policy the target is fixed. With the maturity value guarantee under a variable policy the target is moving and the question immediately arises to what extent should the course of the missile be corrected for every movement of the target. It would seem that soon after launching there is no need for the missile to follow every movement of the target, but as the target is approached then it becomes progressively more important for the missile to reflect faithfully every movement. 1. An accumulation of the premiums charged for the guarantee is the simplest reserve method. The premiums would be accumulated outside the separate account, preferably in fixed interest securities redeemable at the maturity date. On average if the experience corresponds to that expected, then the reserves will be sufficient. However,

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the significant point about the guarantee is that in most years the company will have nil liability on the maturing policies, but in some years this amount will be entirely inadequate. To introduce a margin, the premium accumulated would be larger than the premium charged for the guarantee. 2. A possible approach to the reserves is to consider a prospective method. The actual value of the units attaching to any policy is known at the valuation date. Using simulation or an enumerative or an analytic method, it is possible to obtain the probability density function of the value of the existing units at the maturity of the policy, and similarly the probability density functions of the value at maturity of each of the future premiums can be found. By combining these p.d.f.s and integrating the area under the left-hand tail of the curve bounded by the abscissa and the amount of the maturity value guarantee, the expected cost to the insurer can be found. This amount can then be discounted using mortality and interest and the value of the future premiums for the guarantee deducted to leave the reserve required at the valuation date for the guarantee. This method is attractive from a theoretical viewpoint, but is probably impracticable. However, the method has the advantage that the value of the units actually purchased at the time of valuation assumes an ever-increasing importance as maturity approaches. The reserve basis could be made as conservative as is required for prudence by lowering the mean or increasing the standard deviation of the probability density functions. 3. A radically different approach has been suggested by Benjamin in the U.K. (12). Benjamin argues that the correct way to study maturity value guarantee is to consider what reserves the insurer must hold in order to stay solvent. More precisely, he considers what initial reserves should be set up so that the insurer will have less than a postulated probability, 2%, of becoming insolvent. The initial reserves produced with this approach are predictably very high; for example the initial reserve required could be as high as 25% of the present value of the gross premiums for the entire contract and the question which immediately arises is where is this money to come from. If premiums are increased drastically, then the guarantee would not be attractive from the policyholder's point of view. Alternatively, if surplus is held back to set up these high reserves, then the policyholder should make a contribution for the security he is receiving from these reserves. Although the company

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on average needs to charge only small premiums for this guarantee, it seems that to maintain a probability of ruin of less than, say 2%, requires the office to hold very high reserves. One objection to Benjamin's method is that his results depend greatly on the 'worst' results of a large number of simulations. As Di Paolo says in (10), 'it is the author's opinion that the use of simulated stock market trends to investigate the tail end of a ruin function, precisely because they allow for the random occurrence of such extremes (which in future real life may never occur), will lead to conclusions that may overstate the likelihood of ruin'. Benjamin's study is undoubtedly of fundamental theoretical importance, but the question arises as to how offices should reserve for maturity value guarantees in practice. It is essential for the practical life office actuary to keep the problem in perspective and there are several factors which would appear to mitigate Benjamin's very stringent requirements. (a) If an office wrote only a few term insurance policies, then the reserves it would need to hold to avoid insolvency would be far greater than could be provided by reasonable premiums. However, as more and more lives are insured then the probability of the actual experience deviating from that expected by an amount sufficient to risk insolvency becomes much smaller. Similarly, very high reserves are necessary to ensure solvency when considering the maturity value guarantee under policies maturing in any one year. However, if the office has a good spread of business by year of maturity, then the total reserves which need to be held are much less than the sum of the reserves which would be required for the maturity guarantee in each year considered individually. If the office does not have a good spread of business by year of maturity, then it should reinsure part of the benefits under the maturity value guarantee. It should be remembered that the guarantee applies at only one point in time so that the office will never have to pay out under the guarantee for all policies at the same time, in contrast to an office which guarantees surrender values. (b) It might be convenient for the office to treat its reserves for the guarantee in two parts. First, there are the reserves which form the best estimate of the expected amount required and secondly, there are the additional reserves required to ensure D

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solvency. The latter may be considered as forming part of the general contingency reserves of the office provided that the amount of business with this type of guarantee does not form too large a proportion of the total liabilities. (c) Another factor is that in some cases, but not all, the investment policy of the separate account will be under the control of the life office management. The management could ensure that the separate account is not too volatile through its investment policy, for example by investing a percentage of the assets in fixed interest securities. (d) It should be remembered that it is the actual experience of the separate account which really matters to the life office. No matter what reserve basis is adopted during the term of the policy, the most important point is that the office pays out the claims under the guarantee, if any, as they arise. The ideal reserve basis from the office's point of view should commence at zero and progress smoothly to the ultimate liability although this, of course, is not known until maturity. It is undesirable for the reserves to fluctuate more than the claims under the guarantee. (e) If in real life the stock market did decline by, say, 25% per annum for four or five years, the whole economy of the country would necessarily be adversely affected and lapses would be expected to rise considerably, thus relieving the life office of its liability under the guarantee. (f) The effect of capital gains tax on the separate account is to reduce the volatility of the account and hence reduce the cost of the guarantee. Regulation

The reserve bases laid down by the U.S. state insurance departments for minimum death benefit guarantees under variable annuities are at present being formulated. The state of Maryland has an unfortunate requirement that a reserve of 2% of the total of all gross premiums received should be held irrespective of the value of the units. In other states the reserve basis is as yet not settled. No state appears to have specific regulations on maturity value guarantees. The Canadian Department of Insurance has issued two series of 'Guidelines' to life offices in respect of equity linked insurance and

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annuity contracts with guaranteed benefits. The first set of 'Guidelines' was issued in 1970 and contained a rather crude approach to the reserve problem. A minimum reserve of at least 10% of the guaranteed maturity values was to be held for contracts maturing within one year, even if the value of the units held were twice the guaranteed maturity value; on the other hand, no minimum was specified for maturities in two years' time, regardless of the current value of the units. The revised set of 'Guidelines', dated June 1971, lays down that, 1. The guarantee shall not exceed the sum of the gross premiums paid to the date of maturity. 2. The term of a contract with maturity value guarantee shall not be less than ten years. 3. Amounts payable on surrender of a contract shall not be guaranteed. 4. Where a minimum death benefit guarantee is greater than the amount held for the contract in the separate account, the company shall ascertain the amount at risk and provide in accordance with a method and bases filed with the Superintendent of Insurance for an appropriate reserve in the life insurance fund. 5. For contracts with maturity value guarantee, a 'risk premium' calculated in accordance with a method and bases filed with the Superintendent, shall be charged and allocated to a 'security reserve' within the life insurance fund. The amount of the security reserve at the end of any year shall not be less than the excess of (a) over (b), both with respect to the aggregate of contracts maturing within the following ten years, where, (a) is a special reserve calculated using the formula

where, Gt = total amount of maturity value guarantees under contracts maturing in t years, nt = annual allocations to the separate account under contracts maturing in t years (note that nt does not include the risk premium for the guarantee) and Vt and ät are the special valuation factors set out below: and (b) is the amount held in the separate account in respect of the contracts referred to in (a) together with any

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reserves, other than the security reserve, held in the life fund in respect of the maturity value guarantees under those contracts. Special Valuation Factors n 0-600 1 1-200 1-800 1-200 2 2-990 3 1-180 4-152 4 1-145 5-278 5 1105 6-360 6 1-060 7-395 7 1010 8-375 8 0-955 9-305 9 0-900 10-178 10 0-845 This last requirement suggests that the risk premiums charged for the guarantee be accumulated from the outset and that a contingency reserve, equal to the special reserve as defined in (a) be held for those contracts with 10 years or less to maturity. CONCLUSION

It is hoped that this paper gives an informative account of the past and current development of variable insurance in the United States and that useful comparisons can be made with the United Kingdom. Life insurance in the U.S. is constricted to a very great extent by the regulatory authorities and the U.K. industry should be thankful for the minimal amount of legislation in this country. Many of the controls currently being considered for unit-linked insurance in the U.K. by the Scott Committee are embodied in U.S. legislation. In the author's opinion, it would be regrettable if the Scott Committee recommended differential treatment for unitlinked business compared with conventional life insurance. The special regulation of the U.S. variable annuity should not be emulated. One way of making a clearer distinction between unit-linked insurance and pure investment would be to require, as well as minimum death benefits, a minimum return on the maturity of endowment assurances, and on the vesting of deferred annuities, as is required in Ontario, Canada. Such guarantees appear to be advantageous to both the policyholder and the insurer.

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So long as the currency continues to depreciate in value at the present high rates, it is necessary to consider the best means of providing insurance benefits which will maintain their real value over time. Not enough consideration has been given in the U.K. to life insurance linked to the cost of living. The various designs for variable life insurance in the U.S. do not appear to have any obvious advantages over the unit-linked policies sold in the U.K. The ideal policy might provide death benefits linked to the cost of living while the savings element is invested in equities or property or a managed fund. It would appear very difficult unconditionally to link annuity payments to the cost of living. ACKNOWLEDGEMENTS

The author is very conscious of the honour of being awarded a Winston Churchill Travelling Fellowship and it is the author's hope that the knowledge gained on his Fellowship will be disseminated by means of this paper. The author wishes to express his gratitude to the many actuaries in North America who most generously gave of their time and enabled him to carry out his studies. Further the author wishes to thank his colleagues for their assistance and his employer for the leave of absence granted to him. BIBLIOGRAPHY 1. MACLEAN, J. B. (1956). A new retirement system with provision for variable income. T.F.A. 23, 327. 2. BLUNT, R. H. and LANE, L. H. (1956). Variable annuities. J.S.S. 14, 259. 3. CAMPBELL, P. A. (1969). The Variable Annuity. Published by Connecticut General Life Insurance Company. 4. BRAGG, J. M. and STONECTPHER, D. A. (1970). Life insurance based on the consumer price index. T.S.A. XXII, 333. 5. FRASER, J. C , MILLER, W. N. and STERNHELL, C. A. (1969). Analysis of

6. 7. 8. 9.

basic actuarial theory for fixed premium variable benefit life insurance. T.S.A. XXI, 343. Proceedings of the National Conference on Variable Life Insurance. Variable Life Insurance: Current Issues and Developments. Published by the Insurance Department, University of Pennsylvania (1971). BAILEY, W. A. 'Frequency distributions of stock market price indexes'. Presented at the Society of Actuaries, Seattle meeting, May 1971. KAHN, P. M. (1971) 'Projections of variable life insurance operations'. T.S.A. XXIII, 335. TURNER, S. H. (1969). 'Asset value guarantees under equity-based products'. T.S.A. XXI, 459.

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10. Di PAOLO, F. P. (1969). 'An application of simulated stock market trends to investigate a ruin problem'. T.S.A. XXI, 549. 11. HICKMAN, H. W. (1970). 'A reserve basis for guaranteed benefits under variable annuity contracts'. T.S.A. XXII, 191. 12. BENJAMIN, S. 'A study of maturity guarantees under equity-linked policies'. Presented under the aegis of The Research Committee of the Institute of Actuaries, December 1971. APPENDIX

We shall demonstrate by mathematical induction that the prospective reserve after t years for a fixed premium variable benefit whole life policy issued at age x under the Nylic design is Ft(tVx). We wish to show that Ff(t(Vx) = value of future benefits—value of future premiums. Let value of future benefits and value of future premiums Then we wish to show that for all t, (5) We have

(6) and

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(7)

If w is the limit of life, then since lw = 0, we have from equations (6) and (7), (8) and (9)

Now from equation (2), we have

From equations (8) and (9) it follows that i.e. equation (5) is true for t = w—x—1. Now suppose that equation (5) is true for t = n, so that (10) If we put t = n in equation (2), we have

From equation (10) this becomes

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From equations (6) and (7), we see that this expression reduces to Thus if equation (5) is true for t = n, it is true for t = n—\. But we have proved that the result is true for t = w—x—l. Hence by induction the result is true for all t.