ACTUARIAL COMPARATIVE ANALYSIS OF NATURAL PREMIUM AND LEVEL PREMIUM AND HOW LEVEL

PREMIUM WORKS.BY NWITE SUNDAY C. A RESEARCH STUDENT AND

LECTURER DEPARTMENT OF BANKING AND FINANCE. EBONYI STATE UNIVERSITY – ABAKALIKI.

 

ABSTRACT

 

Insurance contract is a legal contract and because of the

legality, premium is one of the basic consideration for the

acceptance of the insurance risk Canning Vs Farquahar

(1868) stated “NO PREMIUM NO INSURANCE” and Ivamy

(1979) defined insurance as a contract. Based on these, it is

necessary to know how companies determine their premium

charges either on natural method and level premium 

method and it was found that premium under level premium

was better than natural premium and illustrations were

made on the possibilities and recommended that companies

should use level premium method rather than natural

premium.

KEYWORDS

Premium, level premium, natural premium, actuarial

valuation, surrender value, paid up policy.

 

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INTRODUCTION

The contract of insurance is a contract that is based on

utmost good faith and before the contract becomes

enforceable, there must be consideration.

Consideration can therefore be defined as the premium the

insured pays to the insurance company in view of the risk

inured, so that if a loss occur, the insurer will put the insured

in the same financial position he or she was prior to the loss

(Ivamy: 1979) some companies charge level premium, while

others charge natural premium.

HISTORY OF NATURAL PREMIUM.

Natural premium is the situation in life policy where by the

premium charge at the commencement of the contract

continues to increase as the age increases using a mortality

table.

Even a cursory glance at a modern mortality table will reveal

that the chances of dying during any particular year varies

remarkably according to age.

Thus, to take an example, a man who is aged 25 will pay

lower premium, but the premium he is going to pay is higher

as the age increases. The premium must steadily increase as

the age rises, because the risk of death steadily increases

and it must be ensured that each year’s claims are covered

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by each year’s premium. The increase would be sharp until

the time when the premium would become prohibitive.

The position might be modified if it were possible each year

to secure a large influx of younger lives, but in practice this

has never been found to be the case. 

The second difficulty is due to selection; this is the

identification of lives, which from the point of view of

mortality are inferior.

There are two types of methods used to achieve this; one is

by imposing a medical test each year on the participants or

the imposition of the subsequent state of health ignored.

If however, selection is made only at the time of original

entry and there is no medical test each year, the tendency

would naturally be for more of the best and fittest lives than

of the inferior lives to abandon the scheme when the

premium begin to rise sharply, occurring to the greater

chance of death caused by increasing age.

In this case more of the inferior lives would be left which

would lead to more frequent deaths and premiums would

still be further increased in order to cover the claims.

This therefore has made the natural premium system

unworkable and the system almost be completely

abandoned. Many attempts has been made to revive the

scheme or even restrategise it, but all to no avail. This

threaten to the development of an entirely different system,

which is the level premium system.

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The level premium  system, is a system of premium

calculation that stipulates that a single percentage be

collected uniformly throughout the duration of the policy,

This system emphasizes that, if therefore a level premium 

be charged throughout the duration of the policy during a

time of increasing risk, a premium will be payable during the

early years that is higher than is needed to meet the cost of

the risk of a claim. This is in order that there may be

something in hand to meet the cost of the greater risk in

later years when the premium will be less than is required to

cover the risk.

HOW THE LEVEL PREMIUM SYSTEM WORKS

This is going to be illustrated on the assumptions that the

group of whole life assurance is in a closed fund (with no

new entrants once the scheme has started) it may also be

assumed that:

-         There is a large body of new entrants of a given age

(say,25) all of whom have been selected by medical

examination for life assurance.

-         The necessary knowledge is available which will

enable premiums be calculated scientifically.

-         The expenses of ruining the scheme can be ignored.

-         Each policy remains in force until the death of the life

assured, that is none of the policies is surrendered or made

paid-up.

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-         No other circumstances arise which cause any

modification of the plans, and

-         Any margin for safety can be ignored.

In the first year, there will be few deaths causing a moderate

absorption of the premiums; the balance – a very large one –

will go to the reserve.

There are no new entrants because it is a close fund, so that

in the second year there will be slightly fewer premiums

because of the fact that no premium would be collected from

those who died in the first year. The claims will be slightly

greater. The difference between the premiums and the

claims will again go to reserve.

Each year the premium income will be slightly less and the

claims will be slightly more, with the balance still going to

the reserves. The reserve then gradually grows until comes a

time when the premiums balance the claims and there will

be nothing for reserve.

The next year’s claims will slightly exceed premium and the

difference must be drawn from reserve. This reserve then

gradually reduces with every year because more claim will

exceed premiums, until finally when one life is left in. He

pays his last premium and dies, and last premium with the

residue of the reserve is sufficient enough to pay the claim.

This will be so where the assumptions as to interest,

mortality and expenses are exactly those experienced

throughout the whole of the operation.

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FEATURES OF THE LEVEL PREMIUM SYSTEM

The following is a summary of the features of the level

premium system:

-         The total reserve in a closed group of lives (that is,

with no new entrants) increases to a maximum and

then decreases.

-         The reserves for any one particular policy steadily

increases throughout its duration steeply at first and

more gradually later on.

-         The policy period is treated as a whole. Once the

premium is fixed it cannot be altered.

-         The premium must therefore be scientifically fixed.

Knowledge of the probable course of mortality is

required, hence the investigations into the mortality of

the past and the production of mortality tables.

-         Reserves will be invested at interest, so that

knowledge of compound interest is required.

-         Allowance must be made for expenses of

management, commission and a margin for adverse

features.

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-         Also to be noted   in the assessment of premium to be

charged, it is necessary, therefore, to take into

account not only the chance of death at any particular

age but also,

-         The rate of interest which can be earned on reserve if

invested and;

-         The additional amount (called loading) which must be

added to the premium to cover expenses and to

provide a reasonable safety margin.

THE ACTUARIAL COMPARATIVE ANALYSIS OF THE

NATURAL AND LEVEL PREMIUM SYSTEMS

Reserves: consider whole life insurance policy of N1,000

issued to an individual aged 22. In the table below the net

annual premium for this policy is compared with the natural

premiums at various aged of the insured.

                            Net Annual Premium               

Natural

Age                 At Age 22                     Premium

22                    13.28                               2.53

23                    13.28                               2.61

40                    13.28                               4.03

51                    13.28                               12.95

52                    13.28                               13.95

75                    13.28                               86.47

85                    13.28                               189.38

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In the illustration, it is seen that during the early years of the

policy the insured is paying the company more than the

year. By – year cost of the insurance, 13.28 – 2.53 = $10.75

in the first years, and 13,28 –2.61 =$10.67 the second year.

Each excess of annual premium payment offer the cost of

insurance is placed by the company  in a reserve fund which

earns interest at the same rate as that used in computing

the premium. At age 52, the cost of one year of insurance for

the first time exceeds the premium payment. Beginning then

at age 52 and continuing each year there after so long as

the policy is in effect, the company withdraws from the

reserve fund, sufficient to make up the difference 13.95 –

13.28 = $0.69 at age 52 and 86.47 – 13.28 = N73.19 at age

75. The reserve fund on this policy increases throughout the

life of the policy. In accordance with the CSO table used here

the reserve at age 99 would be 1000 v = $975.61 that is the

net single premium for a whole life assurance policy of

N1000 at age 99.

The reserve fund at the end of the year is called the

“terminal reserve” for the policy year. The terminal reserve

less a nominal charge for expenses is called the “cash

surrender value “ of the policy. The insured may borrow at

any time the cash surrender value of his policy without

further collateral and the terminal reserve belongs to the

insured as long as the policy is in force. He could as well

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allow his policy lapse and either take the cash surrender

value or use it to purchase another insurance policy.

MATHEMATICAL ILLUSTRATION FOR LEVEL PREMIUM

PRACTICE.

rv + px ax+z

rv = Ax + r – Px ax+r

= mx+r – mx . Nx+r

   Dx+r     Nx    Dx+r

EXAMPLE:

Find the terminal reserve at the end of the 10th policy year

for an ordinary whole insurance policy of N1000 issued to an

individual aged 22.

100010 V = 1000 A32 - 13.28 a32         

= 100 m32 – 13.28 N32

32                     32

 

1000 m32 – 13.28 N32

           D 32

     

      = 50,165,505

       416,507

     

     = N120.44

From the above, the following conclusions and

recommendations will be made.

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CONCLUSIONS

1.    Natural premium considers the risk yearly.

2.    Situation of the risk may change the policy.

3.    The premium increases as the age increases in natural

premium.

4.    Level premium is the best where the same  premium is

paid.

RECOMMENDATIONS

From this work, the researcher recommended that level

premium is better than natural premium and recommended

that policy holders and insurance companies should consider

level premiums the best option to natural premium

 

REFERENCES

Ayres  F. (1983): Mathematics of Finance, Aslan Student

Edition.

Marshal C. (1989): Insurance of the Person Chartered

Insurance Institute London.

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