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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.

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KEYWORDS
Premium, level premium, natural premium, actuarial valuation,
surrender value, paid up policy.
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.

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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 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.

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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.

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- 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.
- 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

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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.
FEATURES OF THE LEVEL PREMIUM SYSTEM

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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

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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
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

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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 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

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Dx+r Nx Dx+r
EXAMPLE:
th
Find the terminal reserve at the end of the 10 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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