variable annuities and variable life insurance

8/10/2019 Variable annuities and variable life insurance

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V R I B L E N N U I T I E S N D V R I B LE LIF E

I N S U R N C E IN THE U N I T E D S T T E S OF

M E R I C

by

S HAMILTON LECKIE

INTRODUCTION

THE purpose of this paper is to outline the development and currentstatus of variable annuities and variable life insurance in the United

States of America. The author was fortunate in being granted aFellowship by the Winston Churchill Mem orial Trust which enabledhim to travel extensively in North America for two months in thesummer of 1971. It is pointed out that this paper is the result of alarge number of impressions formed by the author and has no claimto be a comprehensive treatise on the subjects. However, it is hopedthat the paper will be of real interest to actuaries and others in theUnited Kingdom.

Part I of the paper deals with variable annuities, Part II with

variable life insurance, and Part III with the special problem ofproviding minimum death benefit guarantees and maturity valueguarantees for these variable products. Variable annuities are muchmore established in the United States than in this country, butvariable life insurance is just in the process of being developed.Mention will be made of the broader issues as well as of actuarialmatters.

ackground information

There are many differences in the conditions under which lifeinsurance companies in the U S operate compared with the U KU.S. life offices are subject to much greater regulation and theregulations are applied by each of the 50 states, plus the Districtof Columbia. Before a life office can write business in any state, itmust be licensed and have its insurance policies approved by thatstate s insurance department. The insurance laws differ in certainrespects from state to state and there are many small offices which

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JSS 20 (2) (1972) 69-112


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S HAMILTON LECKIE

are licensed in only a few states. Agents, that is the salesmen, mustbecome licensed by examination before selling insurance in anystate. The state insurance departments lay down minimum cashsurrender value and valuation bases and also in some cases, theterms 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 thestates also charge a tax of up to 3 on the premium income derivedfrom business written in that state.

Life offices may operate their own mutual funds (U.S. equivalentof unit trusts) and any company writing variable annuities or mutualfund business is also regulated by the Securities and ExchangeCommission (S.E.C.). This federal body regulates anything deemedto be a 'security'. Conventional fixed dollar life insurance andannuities are not regulated as securities, but variable annuities areso regulated and the position of variable life insurance is not yetsettled. The purpose of the S.E.C. is to protect the small investorby, for example, demanding minimum standards of disclosure andby fixing maximum amounts of commission. The S.E.C. is not alwayseasy to contend with and inconsistencies arise in the S.E.C. s dealingswith 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 anew type of policy must expend much time and effort on the regula-tory implications. Only once these constraints have been satisfiedcan the company start to think of the purely actuarial matters suchas setting premium rates. Another major difference is that Americanactuaries are more marketing minded than their British counterpartsand the proprietary companies seem to be more profit-conscious.

PART I THE VARIABLE A NNU ITY

evelopment and regulationThe variable annuity may be defined as an annuity under which

the annuity benefit varies with the investment performance of aseparate account and the annuitant bears the investment risk. Thecontract provides for the annuitant to pay premiums during anaccumulation period and these contributions are invested in aseparate account. The annuity period commences at the end of theaccumulation period and the annuitant thereafter receives periodic


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payments, the cash amount of which depends on the market valueof the separate account at the time of payment. An immediatevariable annuity is the special case with no accumulation period.

The first variable annuity in the United States was the CollegeRetirement Equities Fund C.R.E.F.) system. This was establishedin 1952 as a companion organization to the Teachers Insurance andAnnuity Association, a non-profit-making body providing insuranceonly to teachers. Concern about retirement security had led to acomprehensive study of techniques of combating the inflationaryerosion of pension dollars. The system, which has proved verysuccessful, has been fully described in British actuarial literature byMacLean 1) and by Blunt and Lane 2).

The principle behind the establishment of C.R.E.F. was that avariable annuity would firstly provide protection against inflationby maximizing the amount of funds available at retirement andsecondly, provide an increasing benefit which would not be erodedby the fall in the value of the dollar. Many economists and othershave demonstrated that historically the value of a portfolio ofordinary shares with net income reinvested has, over the long term,increased by more than the cost of living although there is no cor-relation between share prices and the cost of living in the short term .Clearly, however, the yield on a fixed dollar annuity could rise t apoint where future increases in the variable annuity payments arefully discounted.

In the mid-1950s a few small insurance companies were formed tosell variable annuities without having much impact on the industry.However, the S.E.C. became interested in variable annuities beingoffered to the public because of the similarities of the contractto a mutual fund during the accumulation period. After a series ofcourt cases it was established that the variable annuity is a securityand therefore comes under the jurisdiction of the S.E.C. Thisdecision has had far-reaching consequences for the insurancecompanies as it meant that the variable annuity insurer has tocontend with four overlapping spheres of regulatory controls, viz.the S.E.C, the state insurance departments, the state securitycommissions and the Internal Revenue Service. However, theregulatory position has gradually become a little clearer althoughthe variable annuity is still subject to dual regulation as a security

and as an insurance policy. The whole development of the variableannuity has been well documented by Campbell 3).


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The requirements of the S.E.C. are extremely comprehensive andmay be considered in 3 parts viz. regulation of the variable annuitycon tract of the separate account and of selling practices. Thepurpose of the regulations is to ensure that complete disclosure isprovided on the nature of the variable annuity and to preventmisrepresentation in the sale of the product. Before a sale is madethe agent must deliver a prospectus to the purchaser. Prospectusescan be extremely lengthy and contain comprehensive informationon the contract including a description of the expense loadings andthe charges on the separate account the options available the rightsof the policyholder and financial information on the separate

account and on the company. Prospectuses must be approved wordfor word by the S.E.C. and this can be a very long process.

The S.E.C. con trols the maximum amount of commission payableand the level of the expense loadings. Every agent is required topass an examination on the securities business as are specific headoffice personnel such as field supervisors and investment managers.The agent is also strictly controlled on any illustrations used toindicate the possible benefits of the policy. Illustrations can only bebased 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 therequirements for licensing vary from state to state. Agents andassociated head office personnel may be required to take an additionalexamination on variable annuities. The end result of all theserequirements is delay and expense for the company since theregistration with the federal and state authorities may take from 6months to 2 years.

The delays in the development of the variable annuity were not allof external origin. Within the insurance industry itself a majorphilosophical battle waged for a decade as to the desirability ofvariable annuity business. Some companies felt that the policy-holder expects security and certainty from life insurance contractsthat it is the business of insurance companies to insure the investmentrisks and that there would be dangers in the manner of the sellingof these contracts. Furthermore it was argued that the life insuranceindustry should concentrate on using its influence to counter

inflationary pressures in the national economy. However othercompanies insisted that the variable annuity is in fact a sound


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insurance policy which is needed by the public, that inflation isbeyond the control or influence of business, and that the variableannuity would help the purchaser avoid the consequences of inflationthrough participation in the growth of the economy.

Criticism of the variable annuity has now disappeared and mostof the large- and medium-sized companies have entered the variableannuity market but the attitudes of different companies still varyconsiderably. Some companies are very enthusiastic about the vari-able annuity as part of their portfolio of products, but other com-panies only entered the market as a defensive measure for reasonsof competition. The attitude of any particular company is reflectedin its sales force. In some companies 100 of the agents are licensedto sell variable annuities whereas in other companies the proportionis as low as 25 . Also the attractiveness of this business to the agentis affected if the amount of commission paid is less than under theoffice's conventional annuities, either because of the S.E.C. or bydesign.

ypes of variable annuityThere are four main types of variable annuity in the United S tates:

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 introducingthe 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 theI.R.S. and therefore the company pays no tax on the separateaccount backing this type of business. However, non-qualifiedseparate accounts are subject to tax on long-term capital gains.Hence at least 2 separate accounts are desirable if not essential fora company writing both qualified and non-qualified business.

Many pension plans in the U.S. are of the money purchase typeand there are no regulatory restrictions on the relationship betweenthe amount of the pension and final salary. Consequently mostgroup variable annuities are simply a collection of individual policieswith premiums being received in bulk. H.R. 10 and tax shelteredannuities form very attractive business to insurers since there is

B


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tax relief available to the policyholder and the average policy sizeis large. Some companies specialize in this highly competitive marke t.

The individual variable annuity market is not yet fully developed.Here the insurance companies run into direct competition withmutual funds and other savings media and at the same time theagent s commission is much less than for life insurance. On theother hand individual variable annuity contracts are not at a dis-advantage as they are in the U.K. compared with life insurance inthat neither life insurance nor annuity premiums attract tax reliefMany companies have been somewhat disappointed in their effortsto sell individual variable annuities.

A fixed part of an individual non-qualified annuity is regardedas return of capital and this excludable am ount is tax-free in thehands of the recipient. As in the U.K ., the method of taxation resultsin the net benefits under a variable annuity fluctuating less than thegross benefits. If the benefit under a variable annuity is less than theexcludable amount in any year, the excess can be respread overfuture years to increase the tax-free element.

Each of these classes of variable annuity is, of course, availablein fixed dollar form and many companies stress the idea of a balancedannuity in which part of each premium, not necessarily constant,is applied to a fixed dollar policy. The balanced annuity providessome guarantee of dollar amount and protects the benefits in aperiod of deflation. Also the volatility of the annuity benefits islessened but at the cost of a reduction in the equity participation.At the vesting date there is often an option to change the compositionof the balanced annuity. It is also usual to guarantee that at leastthe premiums received to date will be returned in the event of deathduring the deferred period.

Interest rates are currently high in the United States and so theinitial benefit on an immediate variable annuity compares unfavour-ably with the return on a fixed dollar annuity. As in the U.K., fewimmediate variable annuities have been sold. However, the policy-holder is not given the full benefit of the high interest rates in thepremium basis for a deferred fixed dollar annuity and so the variableannuity becomes more attractive the longer the deferred period.

Premium basis

Interest No interest assumption is, of course, made for theaccumulation period. During the annuity period it is not necessary


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to make an interest assumption either, but it is universal practice todo so. Theoretically the accumulated moneys could merely beplaced in the separate account and the expectation of life used todetermine the initial benefit. The initial payment would then bemuch lower than under a fixed dollar annuity although the invest-ment performance of the separate account with regard to both incomeand capital gains would be fully reflected in the subsequent annuityunit values and hence in the annuity benefits.

To overcome the problem of a low initial payout, an interest ratecalled the Assumed Investment Return (A.I.R.) is introduced. Theinitial benefit is obtained by dividing the accumulated moneys bythe appropriate annuity factor based on the A.I.R. The A.I.R.anticipates a certain rate of return in the separate account and theactual investment performance of the separate account is reflectedin the annuity unit value only to the extent that the earned rateof interest exceeds the A.I.R. The A.I.R. thus determines the initialbenefit and also the rate at which the benefits increase for a givenchange in the value of the underlying assets. If the A.I.R . is too highthen there is the possibility that paym ents may no t increase as rapidlyas desired, or may even diminish over the long term. Conversely,if the A.I.R. is too low then the initial benefit will compare un-favourably with the benefit under a fixed dollar annuity. Shouldthe A.I.R. coincide with the dividend yield on the separate account,then the capital gains or losses in the separate account will bedirectly reflected in the annuity unit value. The A.I.R. used in theU.S. ranges from 3 to 5 although some group contracts mayuse up to 6 . The state insurance departments regulate the maximumA.I.R . which may be used and in some states this is only 3 .

Mortality. Mortality assumptions for the variable annuity tend tofollow those for fixed dollar annuities. In the deferred period fairlyconservative mortality guarantees are given using projectionmortality tables. The guarantees are usually arranged so that adeduction of one year in age is made for every complete 15 or 20years by which the year of birth is later than 1900. In the annuityperiod little or no margin is included in the mortality assumptions,unless the group plans are of the experience refund type or theindividual variable annuities are participating; the variable annuitiesoffered by some companies are fully or partly participating withrespect 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 dailyevaluation of unit prices, (b) incurred because of the amount ofregulation and the need to issue a prospectus, and (c) owing toclerical staff being unfamiliar with the variable contract. Notwith-standing these facts, the expense loadings are often taken the sameas for fixed dollar annuities, thus achieving consistency at theexpense of equity. It should be remembered that the S.E.C. will notapprove a contract if the loadings are too high. Sales charges, whichinclude commission, are limited by the S.E.C. to the equivalent of alevel annual charge of 9 , but m any offices pay less commissionthan the maximum. American actuaries are not so concerned withinflation of costs as in Britain and many believe that increasedefficiency and automation will offset the effect of higher salaries andcosts; some U.S. companies have actually reduced the per policyrenewal expenses in the last 10 years.

The mortality and expense guarantees during the deferred periodare usually paid for by an annual charge on the assets. The risk tothe company under these guarantees depends on the amount of theassets at vesting and so the risk can be matched if the reserve forthe guarantees is allowed to accumulate within the separate amount.

apses Premium bases in the U.S. usually include assumptionsas to lapses. The assumptions are fixed with regard to each office'sown experience and account may be taken of the loss (or profit) tothe company in the event of withdrawal in each policy year. Nolapse assumptions are, of course, made for annuities after vesting.

Profit In general, American offices do not expect to make anyprofits from mortality and also expenses are running higher thancharged for. However, a percentage charge is always made on theassets for investment management expenses and usually for theguarantees, and a 1 annual charge on large funds can yield asignificant amount. Few companies, if any, have yet reached thepoint where expenses are being paid for by the direct loading plusthe asset charge. The profit to the company is expected to come fromthe annual charge, and as the assets build up, the charge shouldhelp offset any inflationary increases in renewal expenses.

The U.S. approach to premium rates is highly sophisticated, atleast in the large companies. Once the basic assumptions have beenmade specimen rates can be calculated (this may be regarded as themicroscopic 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 a20- or 30-year future period. The projections incorporate detailedsales forecasts and the emerging cash flow can be calculated byallowing for lapses, amount put to reserve, mortality, etc. The cashflow is then discounted. Stock market performances can also besimulated and some models allow for the lapse rates to vary accord-ing to stock market fluctuations. High lapses combined with un-favourable investment experience in the separate account couldresult in the life insurance company suffering a substantial loss. Onthe other hand, with a favourable persistency experience and goodinvestment results the life office would appear to be able to achievea very attractive return. Mutual companies expect a return of about10 per annum to justify investing current surplus in the develop-ment costs of the variable annuity and with proprietary companiesthe rate may be nearer 15 per annum . Methods of measuringprofitability other than by calculating the return on capital employedare also used. As a result of the projections, the premium rates maybe adjusted, that is the microscopic rates may be adjusted as a resultof macroscopic considerations.

The key to the profitability of the variable annuity lies in theasset charge. The asset charge in fact replaces the excess interestearnings which the insurance companies are accustomed to receivingon fixed-dollar policies. A charge of between and 1 per annumseems small to the average policyholder who tends to pay moreattention to the direct deduction from his premiums. In fact, thissmall percentage can build up to provide a very large income for theinsurer without the insurer taking any of the investment risk. How-ever, the investment performance of the separate account is, ofcourse, reduced by the amount of the charge.

s rv s

During the accumulation period the assets are automatically equalto the liabilities other than for the guarantees given. The minimumdeath benefit guarantee will be considered in Part III. Theoretically,the charges made for the annuity rate guarantees should be accumu-lated right up to the vesting date but often no specific reserves areset up in which case the charges would go directly into surplus.

In the annuity period the reserve in units for each policy is equalto the number of units payable per annum multiplied by theappropriate annuity factor at the then attained age. Some of the


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states set a maximum valuation rate of interest of 3 , but becauseof the mechanics of the variable annuity no valuation strain arisesfrom 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 anannuitant currently aged x with the next payment due in a year'stime.

Let K = dollar amount at the time of valuation of the annualpayment under the annuity.

Let i = A.I.R.,

and suppose the valuation ra te of interest is 3 .If interest at the valuation rate is earned each year in the future,the amount in dollars of the annuity payment one year hence wouldbe

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 beendetermined by dividing the consideration by ax at the A.I.R.(ignoring expenses). Hence no strain arises on purchase if thevariable annuity is valued a t the A .I.R. as though it were a fixeddollar annuity for the same annual amount as the current dollarvalue of the benefit under the variable annuity.

The separate accountThe 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 electedby the policyholders and any change in the investment policy ofthe separate account must be approved by the policyholders. Somecompanies feel that a separate account invested in property wouldbe very attractive but the supervisory authorities would very likelyimpose severe controls on the level of liquidity and on valuationprocedures.

The same investment portfolio can form the assets for bothqualified and non-qualified contracts although theoretically theremight be differences in the investment philosophy since a non-qualified account is taxed on long-term capital gains. However, insome companies at present capital gains are more than offset byoperating losses so that no tax is paid on the separate account.There also would appear to be a case for using different accountsfor the accumulation and annuity periods, since in the latter a majorconsideration should be to try to minimize fluctuations in theannuity payments whereas in the former the only objective is tomaximize the investment return. Some companies use averaging

of the unit price during the annuity period so as to smooth ou t day-to-day fluctuations.

Current situationThe variable annuity can be criticized on the grounds that it does

nothing to guarantee that the proceeds will no t be eroded byinflation. However, no insurer appears to have developed a feasiblemeans of pricing and marketing an annuity with benefits un-conditionally linked to the cost of living. It is, of course, possibleto design an annuity which increases at the rate of inflation butsubject to a maximum rate of, say, 3 per annum. In general thecost of such increasing annuities makes them appear unattractivecompared with fixed dollar annuities. It can be argued that thecurrent high yields on immediate fixed dollar annuities do in factpartly reflect inflation. However, one fact that is quite certain isthat the real value of fixed dollar benefits will decrease every yearthere is inflation; in other words a fixed dollar annuity is effectivelya decreasing annuity in real terms.

The variable annuity is now fairly well established in the UnitedStates and the rate of new developments has slowed down. Some


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companies have not met their initial expectations, but partly thishas been the result of recent uncertainties in the stock market; infact in May of 1971 redem ptions of U.S. mutual funds exceeded salesfor the first time. Another difficulty is that some of the older agentshave been reluctant to sell variable annuities after a lifetime ofpromoting the security of fixed dollar benefits. One principle thathas been clearly demonstrated is that for any office to succeed inselling variable annuities it is essential first to sell the idea to thesales force.

PART I I . VARIABLE LIFE INSURANCEVariable life insurance may be defined as life insurance with benefitswhich vary according to the investment performance of a separateaccount. Premiums may be fixed or variable and there is a widevariety of ways of linking the benefits to the separate account.Before discussing the development of true variable life insurancein the United States, a description will be given of four existingtypes of policy.

Existing types of poli yThe cost of living rider. This is a rider to a whole life or term

insurance policy under which additional insurance may be purchasedeach year without evidence of health for an amount sufficient torestore the real value of the policy. This benefit is not funded, sothat the premium under the policy increases each time additionalinsurance is bought and the risk to the company is limited to anelement of mortality anti-selection.

There are also policies under which the dividends (i.e. the annualbonuses declared on participating business, usually paid in cash)are applied as paid-up additions or as term insurance to the extentof the inflation over the year, before the balance, if any, is paid incash.

Package products. Some U.S. life offices have their own mutualfunds or allow their agents to sell other mutual funds. It is possibleto combine term insurance with systematic investment in the mutualfund. This is similar in effect to the typical United Kingdom unit-linked policy where a small percentage of the premium pays forlife cover and expenses and the balance is used to purchase units ofa unit trust or internal account. However, because of regulatory


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restrictions, two contracts must be issued in the U.S. although onecheque can be used to pay for both contracts. Either the terminsurance or the investment portion can be discontinued withoutaffecting the other. In some offices the investment medium couldalso 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 substantiallyless commission for the agent than does traditional life insurance.Also the investment portion is regulated by the S.E.C. and the boomdays of mutual fund sales may be over. Although unit-linked policiesare proving very successful in the U.K., the Americans are seekingto develop true variable life insurance in which the entire benefitsvary with the separate account investment performance.

quity funding There is available an interesting contract called anEquity Funding Plan under which a lump sum or systematic invest-ment is made in any one of several mutual funds. Insurance premiumsare then loaned to the policyholder using the mutual fund shares ascollateral security. There are strict regulatory requirements as tothe margin between the value of the shares and the accumulatedloan. As well as life insurance, it is possible for other types ofinsurance to be paid for in this way. Clearly to benefit from theEquity Funding Plan the average rate of return on the mutualfund shares must exceed the rate of interest being paid on the loan.

Life insurance linked to the consumer price index. The best exampleof this is a whole life policy with premiums payable throughout lifeas described by Bragg and Stonecipher in a paper to the Society ofActuaries (4). The sum assured under the policy increases each yearby the same amount as the increase in the consumer price index,but premiums remain level. Thus a policy will maintain its realvalue while the policyholder is not required to pay increasedpremiums, so that the inflation risk is borne by the life office. Theonly limit on the increases is that the maximum benefit is twice theinitial 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 ofinflation must be made. In this case the assumption was 3 forthe first five years and 2 thereafter, so that the maximum sumassured is assumed to be reached after 27 years. This select app roachto estimating rates of inflation was made after studying historicalchanges in the cost of living and considering the current situation.


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The company, by assuming the inflation risk, stands to make aloss if inflation runs higher than the estimated rates. Higher ratesof inflation would bring forward the time when the maximum sumassured under any policy is reached and consequently the companywould pay out greater than expected benefits on death within 27years of the commencement of the policy. On the other hand, ifinflation runs at lower levels than predicted, the company stands toprofit. The risk to the company of higher than expected inflationis not significant a t the younger ages; the premiums under the policyare more sensitive to variations in the number of lapses than tovariations with the same estimated probability of occurrence in therate of inflation. Also, it seems feasible that rapid inflation mayencourage persistency and there is the possibility that the higherinflation will be accompanied by higher interest earnings. In the3 years since the policy was introduced, the consumer price indexhas in fact increased by more than 3 per annum.

The provision of cash surrender values was a problem sincenon-forfeiture values must be guaranteed in the U.S. With anindex-linked policy, surrender values can either be fixed at the outsetor expressed per thousand of current sum assured. There was alsoa problem in determining when the Standard Non-Forfeiture Lawwas satisfied. Bragg and Stonecipher decided on cash values andreserves fixed per policy. It was proved that the maximum policyvalue at any point in time occurs if the consumer price index hasnot increased at all up to that time bu t then instantaneously doubles.The surrender values were based on these policy values and as theamounts were greater than produced under the premium basis, thecost 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 thepaid-up amount eventually becomes the minimum death benefitirrespective of the actual course of the consumer price index.

Some of the advantages of the consumer price index-linked policyare:

(a) It gives the cost of living protection directly, rather thanrelying on investment performance to approximate the effectof 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 invest-

ment 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 lifeinsurance cover required by any individual policyholder will changeover the period of one year by an amount which bears little or norelation to the change in the consumer price index. However, thisdoes not absolve the insurer from his responsibility to providebenefits which remain valuable to the policyholder in an inflationaryera. Unfortunately, few companies have shown much interest inindex-linked insurance.

Development of true variable life insuranceIn November 1969, Fraser, Miller and Sternhell presented a

paper to the Society of Actuaries entitled 'Analysis of Basic ActuarialTheory for Fixed Premium Variable Benefit Life Insurance' (5). In thewords of the authors, the paper 'presents an analysis of the basicactuarial theory for life insurance policies which have (i) fixed pre-miums, (ii) the entire reserve held in a separate account, the assetsof which would be invested primarily in common stocks, and (iii)benefits adjusted to reflect the investment performance of theseparate account in such a manner that the policy owners wouldbear the entire investment risk and the life insurance companywould not share any part of the investment risk'.

The method described in the paper relates the benefits to theseparate account in such a way that the reserve per dollar of actualsum assured at the end of each policy year is exactly the same asfor a corresponding fixed benefit policy. The sum assured at theend 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 theadjustment to reflect the fact that a fixed premium is payable, anda Z factor representing the adjustment to reflect the actual investmentreturn on the separate account compared with the Assumed Invest-ment Return (A.I.R.).

The actuarial theory can be developed as follows by consideringa fixed benefit whole life policy with unit sum assured.

The equation connecting successive terminal reserves is

where

(1 ) reserve after r years of a whole life policy issued at

Px = net annual premium for a whole life policy issued atagex .

agex .i = A.I.R.

q x t i = rate of mortality at age x + tl.

Consider a fixed premium variable benefit whole life policy withunit initial sum assured and with the same annual premium as thecorresponding fixed benefit policy. Suppose that the sum assured

of this policy is adjusted as described above. Then we have

where F r = face amount at the end of the r th policy year

2

and i t = actual net investment return on the separate accountover the t th year.

Equation (2) can be expressed as

Applying equation (1) to the right-hand side of this expression wehave

Solving for Ft, we obtain

years


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If we let

and

then

Using this relationship to connect successive sums assured gives thedesired property that the reserve per dollar of actual face amountat the end of each policy year for the fixed premium variable benefitpolicy 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 theAppendix. The theory can be easily applied to other forms of insur-ance. Furthermore, the formula can be extended to policies underwhich the premiums vary in any specified manner, or where thedeath benefit under the corresponding fixed benefit policy varies inany given manner. Note that for a paid-up policy, Y, is unity, andso F t = 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 u = unit value of separate account at the commencement ofthe policy,

UT = unit value of separate account at the end of the t th policyyear.

If the unit values are adjusted to reflect the actual investmentearnings 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 aninitial sum assured of 1 and with an annual premium of Px.

Then Fo= X 0u 0= 1and 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 determinethe change in the number of units of sum assured from the end ofthe (t l)th policy year to the end of the t th policy year. In orderto keep the number of units of sum assured constant each year,the premium payable would have to vary according to the changesin the unit value.

There is complete symmetry between the two approaches. SinceF,_x = X t-1U t-1 equation (4) can be written

i.e.

Also

i.e.

The significance of the Y and Z factors in the basic equationF t = F t 1 Y t Z t

can now be expressed as(a) the role of the Y factor is to adjust the number of units ofsum assured (and hence the sum assured itself) at the beginningof the t th policy year to reflect the fact that a fixed premium of is payable at that time,(b) the role of the Z factor is to adjust the sum assured so as toreflect the change in the unit value over the t th policy year.

x t_1


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V RI BLE NN UITIE S ND LIFE INSUR NCE 87

It should be noted that there is no need to introduce numbers ofunits or unit values and actual sums assured can be determinedsolely from the Y and Z factors.

As may be expected, the Fraser, Miller and Sternhell paperattracted a great deal of attention in the U.S. and this is reflectedin the discussion to the paper. It transpired that various actuarieshad been thinking along similar lines and it seems appropriate tosummarize the reasons why the idea of variable life insurance shouldbecome of more than just theoretical interest in the U.S. at thisparticular time.

he need for v ri ble life insur nce1. Inflation has been part of life in the United States since World

War II . The rate of inflation has averaged about 2 per annumover the last 25 years although the increase in the cost of living in1970 was almost 6 . Consequently, there has been the problem that'good' dollars have been paid for life insurance but the benefits willonly be received in 'devalued' dollars and the public is generallyaware of the erosion in the value of fixed dollar benefits. SomeAmerican economists feel that inflation will markedly decrease asa result of the present Administration's measures, but others arenot so confident and predict that inflation over the next 25 years willcontinue at levels at least as high as in the past.

2. In 1945 the assets of the life insurance companies representedover 50 of all institutional savings but this percentage has fallensteadily to the current level of approximately 18 . Very littleendowment business is now written and the insurance companieshave lost ground to other savings media such as mutual funds, non-insured pension funds, banks and savings and loan associations(U.S. equivalent of building societies). The life companies haveconcentrated on providing pure life protection without providing acomplementary investment service. However, the industry wouldlike to recover some of the lost ground and this has been witnessedin the emergence of variable annuities and in the setting up of m utualfunds by some life companies.

3. Participating policies are very common in the U.S. but thebonus loadings are smaller than in the U.K. at about 10 to 15of the premiums. Also life office investment in ordinary shares islimited 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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S HAMILTON LECKIE

earnings of equities. However legislation has gradually been passedby the various states to exempt separate accounts from the invest-ment restrictions on the general assets.

4. The method of declaring dividends in the U.S. hinges on thedividend formula by which is calculated the amount of dividendfor each policy according to its contribution to surplus in each ofthe three elements of mortality, expense and interest. In recentyears the interest earned by the companies on new money has beenmuch higher than the return on the existing assets, but many lifeoffices use the same earned interest rate in the dividend formulafor all policies regardless of year of commencement and this isinequitable to the newer policies. One of the advantages of using aseparate account approach is that new policyholders can have thefull benefit of the prevailing high investment returns.

Alternative designs of variable life insuranceMany alternative approaches to variable life insurance were

suggested in the discussion to the Fraser, Miller and Sternhellpaper. Basically the excess earnings of the separate account can beused to increase benefits in any desired manner. For example, paid-up benefits or term insurance could be bought with the excessearnings and the additional benefits may be either variable or fixed.The Fraser, Miller and Sternhell design is in fact the particular casewhere the excess earnings are used to increase the sum assured insuch a manner that the policy reserve per unit of sum assured is thesame as for a corresponding fixed benefit policy. For any given timeseries of investment results the pattern of benefits produced bythe various designs differs widely. Six of the more im portant designsare:

1. Dutch this type of policy first appeared in the Netherlandsaround 1953)

This is a fully variable policy under which the premiums, sumassured and reserves are expressed in units rather than dollars anddirectly reflect the relationship between the actual investment per-formance of the separate account and the A.I.R.

2. New York Life Nylic) as described in the Fraser, Miller andSternhell paper)

This is the fixed premium counterpart of the Dutch design under


21/44

V RI BLE NNU ITIES ND LIFE INSUR NCE 89

which the reserve per unit of sum assured, as in the case of the Dutchpolicy, is the same as for a corresponding fixed dollar policy.

3. FairbanksThis is a policy under which a portion of the premium is used to

purchase fixed dollar one year term insurance equal to the excessof the initial sum assured over the amount of reduced paid-upinsurance purchased by the reserve on a corresponding fixed dollarpolicy and the balance of the premium is used to purchase variablepaid-up insurance.

4. WalkerThis is the counterpart of the Fairbanks design where the one-

year term insurance is variable rather than fixed.

5. CooperThis is a buy term and invest the difference policy under which a

portion of the premium is used to purchase fixed dollar one-yearterm insurance equal to the excess of the initial sum assured overthe reserve on a corresponding fixed dollar policy and the balanceof the premium is put on deposit in the separate account.

6. BoothThis is the counterpart of the Cooper design where the one-year

term insurance is variable rather than fixed.

ctuarial formulae

Let

where i t = actual net investment return on the separate accountover the t th year and i = A.I.R.

Sums assured

1. Dutch

2. Nylic

3. Fairbanks


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4. Walker

5. Cooper

6. Booth

eserves

1. Dutch

2. Nylic

3. Fairbanks

4. Walker

5. Cooper

6. Booth

There is, of course, no 'co rrect' design. Each alternative has differentcharacteristics and places different relative emphasis on increasingdeath benefits compared with increasing cash values. Also the extrabenefits can be made fixed or they can remain variable. Some designsare much more volatile than others. The Nylic design, for example,will give a much greater increase in the early years in death benefitsfor a given favourable investment return in the separate accountthan the Walker design, but the converse is true if the separateaccount investment return is inferior to the A.I.R.

egul tionVariable life insurance has to date not been issued in the U.S. and

the principal reason for this is that the regulatory requirements havenot been determined. Traditional life insurance policies are subjectto regulation and variable life insurance will be regulated also. Themajor issues are 'by whom' and 'to what extent' and these problemsare still being resolved.

During 1970 the American Life Convention and the Life InsuranceAssociation of America, whose member offices transact over 99of U.S . life insurance business, set up a join t committee to formulatea united approach to variable life insurance regulation. In Octoberof 1970 this committee made an informal submission to the Securitiesand Exchange Commission on behalf of the life insurance industry.


23/44


The purpose of the submission was to demonstrate to the S.E.C.that variable life insurance is not a security and therefore that theS.E.C. should in no way be involved in the regulation of variablelife insurance. This united approach by the industry to variablelife insurance regulation contrasts markedly with the situation whichoccurred with variable annuities. In that precedent a few companiesstarted selling variable annuities and then were taken to court bythe S.E.C. with the ultimate result that the variable annuity nowsuffers dual regulation.

Specifically the submission proposes that the S.E.C. should notexercise jurisdiction over variable life insurance policies which havefour basic characteristics designed to ensure that the insurance pur-pose and the insurance function of such variable life policies will bepredominant. The four characteristics are:

1. The contract must provide lifetime insurance coverage . Thusendowment and term business are excluded.

2. The contract must be issued for an initial stated amount ofdeath benefit and must guarantee payment of a death benefitof at least equal to such amount. Such a minimum death

benefit guarantee is clearly necessary for the policyholdersprotection.3. The am ount payable upon the death of the insured in any

year must be no less than a minimum multiple of the grosspremium payable in that year by a person who meets standardunderw riting requirements. By excluding policies having veryhigh premiums or a short payment term, this requirementseeks 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 allrequired approvals by state insurance commissioners. Theimplication here is that state insurance regulation is sufficientto protect the consumer.

The S.E.C. responded to the submission by asking for more informa-tion on variable life insurance and this was supplied by the A.L.C.-L.I.A.A. C ommittee in Feb ruary, 1971. At the end of July, 1971 theS.E.C. asked the industry to make a formal submission which willbe followed by a hearing. It now seems likely that no conclusionon the regulatory position will be reached before the middle of 1972.

Full regulation by the S.E.C. would entail the issuance of pros-


24/44


pectuses limitation of charges registration of agents regulationof sales practices and regulation of the separate accounts. Theinsurance companies would of course prefer the S.E.C. not toregulate but this may be an unrealistic hope. However if the S.E.C.does decide to regulate then the industry would press very stronglyfor exemption from the 1940 Investment Company Act. This is theact which limits the amount of commission which can be paid onthe sale of securities. Clearly variable life insurance will not beattractive to the agency force if the maximum commission payableis limited by the S.E.C. to below that payable on conventional lifepolicies.

Some but not all of the states have already modified their insurancelaws so as to permit the issue of variable life. Many of the states whichcurrently permit variable life require disclosure to the buyer ofspecified information about the policy and the issuing companybut not so fully as in a variable annuity prospectus. The questionof whether the state securities departments will become involveddepends on the outcome of the negotiations with the S.E.C. Thereare also unresolved questions about the taxation of variable lifeboth at the company and the individual level. Specifically the trea t-

ment of the investment gains of the separate account in the taxationof the insurance company and the applicability of current taxationrules on the policy proceeds to the policyholder or beneficiary arecomplex issues.

urrent issues

Although most of the industry is very much in favour of theintroduction of variable life insurance there is still a substantialbody of opinion which is opposed to variable life or at least hassome reservations. Many of the arguments which follow were pre-viously voiced at the time when the variable annuity was the subjectof debate.

1. Probably the main objection is to the assumption that variablelife insurance with assets invested primarily in ordinary shareswill provide the best protection against inflation. In the finalanalysis share prices may depend on corporate profits and thereis no guarantee that inflation will be beneficial to profits. Itis argued that the life insurance industry should devote itsresources towards developing policies which are guaranteed to


25/44


maintain their real value. The ultimate in policy design may bea policy under which the death benefits vary with the cost ofliving while the cash value is tied to equity performance, butthis would be almost impossible to achieve in the U.S. becauseof 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 methodsof selling variable life which can be used by the agent, especiallyas the public will not understand how benefits are related to theperformance of the separate account. Clearly the policyholdermust appreciate the volatility of the benefits and that in contrastto conventional policies, surrender values are not guaranteed asto dollar amount. It is perhaps unfortunate that proposals forself-regulation are not in the American tradition, but theprevailing U.S. attitude is that self-imposed constraints arean infringement on their freedom of action and that only thelaw need be obeyed.

3. With conventional life insurance, the principal source of profitsand surplus to the life office is the excess interest earnings.However, with variable life, superior investment results arepassed directly to the policyholder and it may be asked whythe proprietary companies should offer variable life if theycannot 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 seemspossible that the introduction of variable life insurance will notbenefit the smaller companies, some of which will not generatesufficient business to recoup their development costs.

4. There are fears about the effect on the economy of variable lifeinsurance. At present only estimates can be made of the volumeof business which will be written, but it is conceivable thatvariable life could form 50 of all new whole life businesswithin a few years. Large-scale investment in ordinary sharesby the insurance companies might result in a shortage of stockwhich would push prices artificially high and disturb thenation's capital market. At the same time, the life insurancecompanies would reduce their holdings of fixed interest securitiesand cut back on their supply of mortgage money with theresult that interest rates could rise considerably.In reply to this reasoning, it is pointed out that variable


26/44


annuities have had no noticeable effect on the economy, al-though the predictions on the impact of variable life are al-together on a much larger scale. Almost certainly, variable lifewill have some effect on the economy, but the magnitude ofthe effect can only be guessed at. In the event it may be thatthe investment patterns of the life insurance companies willchange so gradually that any effect on investment patternswill not be detrimental.

In general, it may be said that most of the arguments against variablelife insurance are made out of concern for the industry as a whole.However, each individual office will make its decision on whetheror not to introduce variable life on the basis of its own interests.

Current situationThe current position is that at least three of the large U.S. com-

panies are well advanced in their plans to market variable life insur-ance. Almost every other company is following developments avidlywithout proceeding too far before the outcome of the negotiationswith the S.E.C. An excellent review of the development of variablelife insurance is given in ariable Life Insurance Current Issues andDevelopments 6). The three designs of variable life which willcertainly be sold are the Nylic design, the Dutch design and theWalker design. One company has had its policy documents for theNylic and Dutch designs approved by several of the state insurancedepartments.

It is interesting to compare the three main designs. Most companiesare assuming that fixed premiums are a prerequisite of variable lifeinsurance but this may not be true. The Dutch design is the onlyone with variable premiums and this has several interesting con-sequences. If the separate account performs well, then the policy-holder will be required to pay a larger premium and if the accountperforms badly then a smaller premium will be asked for. This mayhave a beneficial effect on lapses, but the design contravenes theidea of price cost averaging. From the company viewpoint, favour-able investment performance will result in a larger expense loadingbeing received, and vice versa. The variable premium design willalso affect the cost of the minimum death benefit guarantee.

The Nylic design is more volatile than the Walker design and sogood investment performance will be more obviously reflected in


27/44


the sum assured. However, since the sum assured under the Nylicdesign depends on both the Y and Z factors, it is possible for theZ factor to be greater than unity but the sum assured to decreaseover the period of one year. I t may be difficult to explain to the policy-holder why his sum assured has decreased while the separate accountearnings were greater than the A.I.R. The Walker design is akin to aparticipating policy in which dividends are declared in paid-up form,the main difference being that negative paid-up additions are inevit-able in some years.

The pricing of variable life insurance cannot be finalized until it isknown whether the S.E.C. plans to regulate, particularly with regardto commission. However, it is anticipated that variable life premiumswill be marginally dearer than for comparable fixed benefit policies.The A.I.R . is likely to be 3 and variable life policies offered by themutual companies will probably be participating in the mortalityand expense elements.

Mortality assumptions will be the same as for fixed benefitpolicies. However, one difference with variable life is that favourableinvestment 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 atrisk cannot be pre-determined and have the potential of increasinggreatly. The expenses of variable life will be higher than normalbecause of the development costs and the increased running costsof the separate account. To operate variable life successfully, it willbe almost essential to be highly computerized, which may not beeasy for small companies. Again, the profitability to the companyof variable life will depend mainly on the asset charge.

The development of variable life insurance is currently in a mostinteresting phase. No further steps can be taken to issue the firstpolicy until the negotiations with the S.E.C. are resolved, but mostoffices are taking advantage of the hiatus to examine every aspect ofvariable life, even if no commitment has been made to proceed withthe development of it.

The marketing impact of variable life can only be guessed at. Everyshade of opinion can be found, from high hopes that this will be thegreatest innovation for decades, to predictions of dismal failure. Oneof the principal unknowns is the extent to which sales of variable lifewill represent a real increase in new business rather than just aswitch from conventional life insurance. Since the initial sum assured


28/44


will be guaranteed on death and the premiums will be almost thesame as for conventional life a policyholder would seem to haveeverything to gain and nothing to lose from a variable life policy.Although surrender values will not be guaranteed in a variable lifecontract there is the possibility of substantially improving on theguaranteed surrender values in a comparable fixed dollar policy.On the other hand it will be difficult to explain to a policyholderexactly how the benefits vary. Part of the appeal of the typicalU.K. unit-linked policy is the very simplicity of the contract butthis will not be so with variable life insurance. Clearly the agents willhave to be carefully trained on the correct means of selling variablelife policies.

PART I I I MINIMUM DEATH BENEFIT GU ARA NTEE SA N D M AT U R I T Y VA L U E G U A R A N T E E S

The basic principle behind variable annuities and variable lifeinsurance is that the policyholder assumes the investment risk.Nevertheless it seems desirable that the insurer should provide someguarantee as to the benefits which will be paid. Guarantees on deathor maturity would seem to be of real value to the assured so longas they can be provided at reasonable cost. At the same time theinsurance company by undertaking a financial risk will receive apremium for doing so which gives the insurer additional opportunityfor profit providing the correct premiums are charged. It is betterfor the insurance company to increase premium income rather thanto reduce liabilities. However there are many unresolved questionsabout these guarantees particularly about the maturity valueguarantee.

urrent situation

Most variable annuities in the U.S. provide sometimes optionallythat on death during the deferred period a return will be made ofthe premiums paid to date or the value of the units whichever ishigher. The guarantees may be attached to both single premiumand systematic policies and usually the benefit ceases at age 65 oron earlier vesting. No office appears to be offering any form ofguarantee of the value on vesting or earlier surrender with variableannu ities although a guaranteed cash option at vesting of allpremiums paid to date would seem to be an attractive selling point.


29/44


All U.S. variable life insurance contracts will incorporate aminimum return on death of the initial sum assured; this is one ofthe conditions laid down in the submission to the S.E.C. No assetvalue guarantees in variable life policies are at present contemplated,but again there is scope for including a guaranteed surrender valueat, say, age 65. It is possible that if variable life is a success, the trendwill be to introduce variable endowment policies, perhaps with amaturity value guarantee. One insurance company does provide amaturity guarantee in conjunction with certain mutual funds, butthe response by the public has been weak, mainly because thecharge for the guarantee, at 6 of all money invested, seems veryhigh.

In Canada the situation with respect to variable insurance ismore akin to the U.K. than to the U.S. Equity-linked endowmentand deferred annuity policies which provide guarantees on deathand maturity are commonly sold. In 1970 the Ontario SecuritiesCommission indicated that certain types of equity-linked productswould come under the supervision of that body if they did notcontain benefits at death or maturity that would at least equal 75of the gross premiums payable. The securities commissions in someof the other provinces are moving towards similar requirementsand most companies have chosen to include 75 or 100 guaranteesin their products.

he problemsWith both types of guarantee the twin problems are the calculation

of premiums and the determination of suitable reserves. Tradition-ally, actuaries have always considered mean values in their assump-

tions, but with these types of guarantee, it is necessary to considerthe whole range of possible outcomes of investment performanceand attach probabilities to each value. The questions of premiumsand reserves may be considered from three different viewpoints.

heoryIt is quite safe to say that not nearly enough theoretical work has

been done on the nature of these guarantees, which provide somehighly intriguing problems. The techniques which are appropriatesuch as mathem atical statistics, risk theory and computer simulation,are not generally familiar to most actuaries.


30/44


egulationThe only concern of the regulatory authorities is to ensure solvency

for any company offering these guarantees.

ffice practiceThe viewpoint of the offices writing these guarantees is that pre-

miums should be adequate but competitive. Reserves also shouldbe adequate, but at the same time the company does not want toover-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 guarantee1. The simplest approach to calculating premiums for the mini-

mum death benefit guarantee under a variable annuity is to assumetha t the separate account un it values will grow at a constant rate of,say, 5 per annum . The value of the units attaching to any policyis thus known for each policy year and the excess of the gross pre-miums over this value is the estimated liability to the company ondeath. The cost of the benefit can be calculated for any age at entryand single or annual premiums obtained. The premium is usuallyexpressed as a percentage of the office premium. This method takesno account of fluctuations in the unit value, but is certainly veryconvenient. However, the m ethod is useless for variable life insurance,since an assumed rate of growth higher than the A.I.R. implies thatno payments would be made under the guarantee.

2. Perhaps the commonest method by which the premiums for theguarantee have been calculated for variable annuities is using ahistorical approach. A representative ordinary share index over along period of time is chosen and adjusted as appropriate to reflectthe inclusion of net reinvested income, capital gains tax and thecharge against the assets. The net allocation to the separate accountunder any policy is then assumed to be invested at these prices andsimilarly the value of the policy in every subsequent year can befound. For each year of entry, a series can be formed of the amountwhich the company would have paid out under the guarantee. Hencenet premiums can be calculated for each year of commencement.

The office premium can be taken as the average of the historicalnet 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 simulationto obtain series of stock market prices. Simulation may be definedas a process which gives possible outcomes of an event by applyinggiven data in a random manner. Using as data the changes which haveoccurred from year to year or from month to month in an adjustedordinary share index, the value of an investment after a period oftime can be simulated any number of times and a frequency dis-tribution built up. A series can then be obtained of the amountpayable under the guarantee and the benefit costed as before. Thismethod is appealing from a theoretical viewpoint and may beregarded as producing a premium free from the irregularities ofactual history. M ost U .S. companies are intending to use simulationin pricing the minimum death benefit guarantee for variable lifeinsurance.

An alternative approach to simulation has recently been putforward by Bailey (7). The approach is one of enumeration ratherthan simulation and does not involve the generation of randomnumbers. Bailey uses a computer program named 'Dice' whichcalculates the complete frequency distribution of the expected sumsof the faces which turn up when any finite number of dice are ro lled;each die can have any number of faces, each die can have anyamount 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 completefrequency distribution of the value of an investment of 1, either madeonce only or systematically, can be obtained using this statisticaltechnique.

4. It is possible to calculate the premiums for a minimum deathbenefit guarantee by representing stock market fluctuations as amathematical function and applying risk theory. Kahn (8) hasrecently suggested tha t stock m arket price changes may be approxi-mately represented by a log-normal curve. In his paper Kahn com-pares the single premiums for the minimum death benefit guaranteeunder a paid-up whole life policy obtained by simulation and bythis analytic method. The advantages of the analytic approach arethat it provides more precise measurements of the fluctuations ofthe stock market and that it is less expensive than simulation.

It is salutary to examine the nature of the assumption made withthe 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 per-formance of a particular separate account over the next 50 years. Acomparison may be made with the early insurance companies whichhad to use population mortality statistics when setting premiumrates. Fortunately at that time mortality was generally improvingand the offices could exercise selection over the lives to be insured.

It would seem to be highly improbable that the factors whichdetermine investment returns will produce, over the next 50 years,results bearing the same characteristics as in the past. However, thefirst stage is certainly to examine past history, and this can only bedone by looking at a suitable general index. The next step shouldbe to adjust the distribution of investment results so as to fit inwith the expected characteristics and the investment philosophy ofany particular separate account. For example a property fund isgenerally regarded as being much less volatile than an equity fund,although this may not be true if the fund is concentrated in com-paratively few individual properties. However, it will be very manyyears before anything other than one of the standard stock indicescan be considered suitable.

remiums for m turity value guarantee1. To estimate the value of the net allocations to a separate

account in n years time, an average growth rate could be chosenand the net allocations accumulated at this rate. However, thisapproach is totally inadequate since any reasonable growth ratewill result in an estimated maturity value higher than the guaran-tee, implying that there is no liability. What is required is thedistribution of possible maturity values, not just the mean.

2. As with the minimum death benefit guarantee, an historicalapproach is possible. Using a general stock index, adjusted as neces-sary, it is straightforward to calculate the maturity value for a policyof any term which commenced in any past year. Hence the amount,if any, which the company would have had to have paid out can becalculated. The average of these amounts can be divided by xn togive the average net annual premium. Since the nature of theseguarantees is not yet fully understood, it is prudent to include alarge contingency loading in the office premium.

It is sometimes said that a maturity value guarantee is purely afinancial risk, whereas a minimum death benefit guarantee combinesfinancial and mortality risks. This is not really true, since the


33/44


maturity value guarantee is only paid out on surviv l to maturity.However, the minimum death benefit guarantee is much moresensitive to a change in mortality rates than is the maturity valueguarantee. Thus premiums for a minimum death benefit guaranteevary much more by age than the premiums for a maturity valueguarantee. The cost of the latter guarantee is, however, markedlyreduced by lapses.

3. The premiums can be calculated by developing the statisticaldistribution of the maturity value of a policy. This is the approachused by Turner in his prize-winning paper 'Asset Value Guaranteesunder Equity-Based Products' (9). Turner's paper considers threeareas:

(a) analysis of the probability density function of rates of returnon ordinary shares,

(b) presentation of a general simulation model for evaluating thenet 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 thenet risk premium to a change in the term of the policy, in the meanrate of return on the separate account and in other factors, can beeasily appreciated.

4. An alternative approach was put forward by Di Paolo in 'AnApplication of Simulated Stock Market Trends to Investigate aRuin Problem' (10). An investment risk premium for the guaranteeis 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 businessis calculated. The risk fund in this case is the accumulation of thepremiums received for the m aturity value guarantee less the amountspaid out under the guarantee, together with mortality profits lesslosses from the minimum death benefit guarantee. Using simulation,a distribution function of the risk fund can be established and hencethe probability of ruin calculated. Di Paolo regards an investmentrisk premium as adequate if the probability of the risk fund beingin a state of ruin after the last contract matures is less than about10 .

eserves for minimum de th benefit guarantee1. The most obvious method of reserving for this benefit is to

accumulate premiums received less benefits paid out. This method


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2 S HAMILTON LECKIE

is simple, but hardly adequate as the reserve could become negativeand no account is taken of future benefits.

2. A one-year term insurance reserve could be held for an amountat risk, if positive, of the guaranteed benefit less, say, 75 of thecurrent value of the units. This method probably gives adequatereserves overall. It is extremely unlikely that the separate accountunit value will drop sufficiently over the period of one year to give anaverage fall of 25 , but on the other hand, no account is taken ofbenefits in future years. This omission is more important for single-premium contracts than for systematic policies, where the implicitassumption is that the future premiums for the guarantee will beat least sufficient for the future benefits.

The method also suffers from the disadvantages that an initialstrain is inevitable and that the reserve can fluctuate greatly fromyear to year. However, it should be borne in mind that the amountof the reserves for the guarantee may be relatively insignificant.

3. In 1970 Hickman presented a paper to the Society of Actuariesentitled 'A Reserve Basis for Guaranteed Benefits under VariableAnnuity Contracts' (11) Hickman's paper reflects the belief thatmost of the 51 jurisdictions in the United States would be unwillingat this time to base reserves on the result of simulations, but wouldinstead require an approach more similar to traditional methods.His approach is deterministic rather than probabilistic, not becauseit is theoretically better, but because it may be more acceptable tothe state insurance departments. Reserves for both minimum deathbenefit and maturity value guarantees are considered by introducingchange factors, which are analogous to the interest factor in normallife insurance reserves. Applying a change factor n to the currentunit value gives the expected unit value after n years on a con-servative basis. In the determination of what specific change factorsshould be assumed for reserve purposes, w-year change factors werecalculated from historical data for values of n between 1 and 40.The change factors chosen should be smooth to avoid discontinuitiesin reserve requirements, and the factors should be conservative soas to provide adequate reserves. The factors were chosen as

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


35/44


are compared with the lowest historical result the 10th percentilethe 20th percentile and the median values. Fn is consistently moreconservative than the 20th percentile and at durations 1-2 andover 20 is more conservative than even the 10th percentile.

Hickman recognizes the disadvantages of both retrospective andprospective approaches; the former fails in that no account is takenof the current investment situation or of the future benefits whereasthe latter produces reserves which depend on the unit value at thedate of valuation and which can therefore fluctuate considerably.An average reserve is suggested which moves one-fifth of the wayfrom the accumulation of the previous reserve towards the newprospective reserve thus smoothing out the fluctua tions to someextent. However even with this modification the prospective methodwould historically have resulted in fluctuations in the reserve greaterthan the fluctuations in the claims arising under the guarantee.

eserves for m turity value guaranteeThis is a more difficult problem than the reserves for the minimum

death benefit guarantee since the entire liability to the office isunknown until the very end of the contract. With a non-profitendowment policy the insurer knows that he must pay out say1 000 at the end of the term. The two end-points of the reserve curveare known and only the precise shape of the curve remains to bedetermined by the actuary. However with a maturity value guaranteeonly the starting point is fixed and the ultimate liability may be zeroor it may be several hundred pounds. A colourful analogy may bedrawn with a guided missile aiming at a target. With a conventionalendowment policy the target is fixed. With the m aturity value guaran-tee under a variable policy the target is moving and the question

immediately arises to what extent should the course of the missilebe corrected for every movement of the target. It would seem thatsoon after launching there is no need for the missile to followevery movement of the target but as the target is approached thenit becomes progressively more important for the missile to reflectfaithfully every movement.

1. An accumulation of the premiums charged for the guaranteeis the simplest reserve method. The premiums would be accumulatedoutside the separate account preferably in fixed interest securitiesredeemable at the maturity date. On average if the experience corre-sponds 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 thecompany will have nil liability on the maturing policies, but in someyears this amount will be entirely inadequate. To introduce a margin,the premium accumulated would be larger than the premium chargedfor the guarantee.

2. A possible approach to the reserves is to consider a prospectivemethod. The actual value of the units attaching to any policy isknown at the valuation date. Using simulation or an enumerativeor an analytic method, it is possible to obtain the probability densityfunction of the value of the existing units at the maturity of thepolicy, and similarly the probability density functions of the valueat maturity of each of the future premiums can be found. By com-bining these p.d.f.s and integrating the area under the left-hand tailof the curve bounded by the abscissa and the am ount of the maturityvalue guarantee, the expected cost to the insurer can be found. Thisamount can then be discounted using mortality and interest and thevalue of the future premiums for the guarantee deducted to leave thereserve required at the valuation date for the guarantee.

This method is attractive from a theoretical viewpoint, but isprobably impracticable. However, the method has the advantage thatthe value of the units actually purchased at the time of valuationassumes an ever-increasing importance as maturity approaches. Thereserve basis could be made as conservative as is required forprudence by lowering the mean or increasing the standard deviationof the probability density functions.

3. A radically different approach has been suggested by Benjaminin the U.K. (12). Benjamin argues that the correct way to studymaturity value guarantee is to consider what reserves the insurermust hold in order to stay solvent. More precisely, he considerswhat initial reserves should be set up so that the insurer will haveless than a postulated probability, 2 , of becoming insolvent. Theinitial reserves produced with this approach are predictably veryhigh; for example the initial reserve required could be as high as25 of the present value of the gross premiums for the entire contractand the question which immediately arises is where is this moneyto come from. If premiums are increased drastically, then theguarantee would not be attractive from the policyholder's point ofview. Alternatively, if surplus is held back to set up these high

reserves, then the policyholder should make a contribution for thesecurity he is receiving from these reserves. Although the company


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V RI BLE NNUITIES ND LIFE INSUR NCE

on ver ge 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 dependgreatly on the 'worst' results of a large number of simulations. As DiPaolo says in (10), 'it is the au tho r's opinion that the use of simulatedstock market trends to investigate the tail end of a ruin function,precisely because they allow for the random occurrence of suchextremes (which in future real life may never occur), will lead toconclusions that may overstate the likelihood of ruin'.

Benjamin's study is undoubtedly of fundamental theoreticalimportance, but the question arises as to how offices should reservefor maturity value guarantees in practice. It is essential for thepractical life office actuary to keep the problem in perspective andthere are several factors which would appear to mitigate Benjamin'svery stringent requirements.

(a) If an office wrote only a few term insurance policies, then thereserves it would need to hold to avoid insolvency would be fargreater than could be provided by reasonable premiums.However, as more and more lives are insured then the pro-bability of the actual experience deviating from that expectedby an amount sufficient to risk insolvency becomes muchsmaller. Similarly, very high reserves are necessary to ensuresolvency when considering the maturity value guarantee underpolicies maturing in any one year. However, if the officehas a good spread of business by year of maturity, then thetotal reserves which need to be held are much less than thesum of the reserves which would be required for the maturityguarantee in each year considered individually. If the officedoes not have a good spread of business by year of maturity,then it should reinsure part of the benefits under the maturityvalue guarantee. It should be remembered that the guaranteeapplies at only one point in time so that the office will neverhave to pay ou t under the guarantee for all policies at the sametime, in contrast to an office which guarantees surrender values.

b) It might be convenient for the office to treat its reserves forthe guarantee in two parts. First, there are the reserves whichform the best estimate of the expected amount required andsecondly, there are the additional reserves required to ensure

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solvency. The latter may be considered as forming part of thegeneral contingency reserves of the office provided that theamount of business with this type of guarantee does not formtoo large a proportion of the total liabilities.

(c) Another factor is that in some cases, but not all, the invest-ment policy of the separate account will be under the controlof the life office management. The management could ensurethat the separate account is not too volatile through its invest-ment policy, for example by investing a percentage of theassets in fixed interest securities.

d It should be remembered that it is the actual experience ofthe separate account which really matters to the life office.No matter what reserve basis is adopted during the term of thepolicy, the most important point is that the office pays outthe claims under the guarantee, if any, as they arise. The idealreserve basis from the office's point of view should commenceat zero and progress smoothly to the ultimate liability althoughthis, of course, is not known until maturity. It is undesirablefor the reserves to fluctuate more than the claims under theguarantee.

(e) If in real life the stock market did decline by, say, 25 perannum for four or five years, the whole economy of the countrywould necessarily be adversely affected and lapses would beexpected to rise considerably, thus relieving the life office ofits liability under the guarantee.

(f) The effect of capital gains tax on the separate account is toreduce the volatility of the account and hence reduce the costof the guarantee.

RegulationThe reserve bases laid down by the U.S. state insurance depart-

ments for minimum death benefit guarantees under variable annuitiesare at present being formulated. The state of Maryland has anunfortunate requirement that a reserve of 2 of the tota l of allgross premiums received should be held irrespective of the value ofthe units. In other states the reserve basis is as yet not settled. Nostate appears to have specific regulations on maturity value guaran-tees.

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 'Guide-lines' was issued in 1970 and contained a rather crude approach tothe reserve problem . A minimum reserve of at least 10 of theguaranteed maturity values was to be held for contracts maturingwithin one year, even if the value of the units held were twice theguaranteed maturity value; on the other hand, no minimum wasspecified for maturities in two years' time, regardless of the currentvalue of the units.

The revised set of 'Guidelines', dated June 1971, lays down that,

1. The guarantee shall no t exceed the sum of the gross premiums

paid to the date of maturity.2. The term of a contract with ma

variable annuities and variable life insurance

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