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Premium is the price for insurance.Consideration paid by the policy holder tosecure risk cover.

Can be paid in lump sum as Single premium orin instalments at regular intervals as specifiedin the contract.Always payable in advance.Non-payment of premium may endanger thecontinuance of the policy.

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Elements considered in building premiums:

1. Mortality2. Interest3. Expenses

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Census mortality is not used to find themortality experience.

Mortality experience of the insured livesobserved in the recent past is used as a basisfor estimating the probabilities of survivals

and deaths.If out of 10,000 lives all aged 35, 18 diewithin 1 year i.e., before attaining age 36, the

observed mortality rate is 18 / 10,000 =0.0018.

This is denoted by qx.

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The observed rates of mortality, from certainage till extreme age is subjected to a process

of graduation.The process of graduation enables to findthe limiting values of mortality rate ( q x ),when the data is increased infinitely.

These graduated values are used forconstructing mortality tables which containmortality functions (viz) lx , dx , qx , px , Lx , Tx and ex for successive ages x.

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lx : Denotes the number of persons livingwho are at age x out of the l0 persons born.

These persons are also those who survivedone year out of the lx-1 persons.

d x: Denotes the number of persons dyingbetween ages x and x+1. d

x = l

x-l

x+1.

q x: Denotes the probability that the personaged x dies within 1 year (i.e.,) beforereaching age x+1.

q x is also called the rate of mortality at age x.

qx = dx / l x

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p x: Denotes for successive ages, theprobability that a life aged x survives to agex+1.

px = lx+1/ l x or px = 1 - qx

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npx : The probability that a person aged xsurvives „n years.

npx = (lx+n)/l x

lmqx: The probability that a person aged xdies within the next „m years.

lmqx = (lx-lx+m) / l x= 1 -mpx

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mlqx: The probability that a personaged x will die between the age x+m

and age x+m+1mlqx = (dx+m)/l x = (mpx) * (qx+m)

mlnqx : The probability that a personaged x dies within n years following myears from now i.e., between the yearsof x+m and age x+m+n

mlnqx = (lx+m – lx+m+n)/l x= mpx – m+npx= mpx * ln qx+m

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Mortality tables giving rates depending onboth age and duration, i.e., time elapsedsince entry, are called Select MortalityTables.

The process of accepting only such lives as

coming up to the desired standards is called„selection .

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The rate of Mortality of a person who has just now

entered at age 36 is denoted by q [36].

The rate of Mortality of a person now aged 36 butentered one year ago at age 35 is denoted by q [35]+1 ; the figure 35 inside brackets denoting the age atentry and the figure 1 outside brackets denoting theduration since entry, the sum of these two figuresgiving the age attained.

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In constructing select rates of Mortality, therefore

each group of lives entering at a particular age say(x) should be considered separately and a separateMortality Table should be prepared giving:

q[x] q[x]+1 q[x]+2 q[x]+3 etc………

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The effect of initial selection wears outrapidly and after a certain stage thefactor of duration ceases to have anysignificance and mortality thereafter

depends only on the age attainedirrespective of the duration, sinceentry.

The period, on and after which theduration since entry ceases to have anyeffect on the rate of mortality for agiven age, is called the „Period OfSelection .

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A mortality table in which the rates in theselect period are omitted and only theultimate rates are tabulated is called anUltimate Mortality Table, which are usedfor valuation purposes.

Normally ultimate rates are higher thanselect rates for the same attained age asduration elapses, the effect of selection isworn out.

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Insurance works on the law of largenumbers.Consider 1000 persons all aged 40 areinsured for Rs 10,000 each for 1 year.

If the rate of mortality is 0.003 at age 40 thenexpected claims = 1000 * 0.003 = 3.

Claim payable = 3*10,000 = RS 30,000.Premiums = 30,000/1000 = Rs.30 to eachperson.

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Since the rate of mortality at age 40 isassumed to be 0.003, the probability that aperson aged 40 will die within a year is0.003.

Amount of premium chargeable= 0.003 * 10000 = Rs 30/= each.

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Consider a group of 10,000 persons aged 35seeking a cover of Rs 10,000 for next 10 years.

Let us say expected No of deaths among 10,000all aged 35 before attaining age 45 is 462.

44 ∑ dx = 462 35

Claims payable = 462*10000=46,20,000Single premium = 46,20,000 / 10,000 = Rs.462

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44

∑ dx 35

------------ = (462 / 10,000)*10,000 = 462

I35

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Let us assume insurer earns interest @ 6%.

PV (Present Value) of claim payable is 32,38,922The amount to be shared by 10,000 persons is

32,38,922 / 10,000 = 323.892

This single premium is equal to PV of benefits tobe provided under the plan.

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The probability that a person aged 35 yrs willdie within a year is d 35/l 35 (assume d35=28)

Premium ignoring interest = SA * PROB10,000*(28/10,000) = 28

Premium taking care of interest= 10,000*(28/10,000) * 6%= 2,64,152 / 10,000 =26.4152

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Premium for second year = SA * PROB ofdeath in second year * discounting factor

for 2 yrs. Assuming d x =31, 10,000*d36/l 35 * v2

10,000*(31/10,000) * 0.89 = 27.5900

Premium for third year = SA * PROB ofdeath in third year * discounting factor for 3

yrs.

Assuming d x =34, 10,000*d37/l 35 * v3

10,000*(34/10,000)*0.83962 = 28.5471

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For remaining years up to 10 years:

2,64,152 + 2,75,900 + 2,85,471 +…..to 10 terms = --------------------------------------------------------

10,000=32,38,922 / 10,000 = 323.892

If the period of insurance is extended to theentire life span it is a Whole Life Insurance.

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When the benefit is payable only at the endof the selected period provided the lifeassured is alive then, it is called PureEndowment Assurance.

Ex: If 10,000 persons aged 35 take thisassurance they will get Rs 10,000 at the endof 10 years if they are alive then

Premium ignoring interest =9,53,80,000 / 10000 = Rs.9538

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Premium with interest @ 6% =(95380000)v10 = (95380000)*(0.55839)

= 5,32,59,238/-

Single premium / person = 5,32,59,238/10,000= Rs.5326/-

Probability approach = l45/l 35 = 9538/10,000(10,000)*l45/l 35=10,000*(9538 / 10,000) = Rs.9538/-

If Interest @6% is accounted then10,000 * (9538/10,000)*(0.55839) = Rs.5326/-

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Natural premiums can be replaced bylevel premiums throughout the term ofinsurance.

Insured pays little more premium thanwhat is required in the earlier durations.Excess premium paid during the earlierdurations would meet the additionalcost of insurance at higher ages, later.Reserves under each life insurance policywould be generated because of this level

premium concept.

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At younger ages insured pays a little morethan required.

This additional payment with interestover a period of time would go a longway to meet the cost of benefits at higherages.

Easy to make provisions for payments,and makes it easy for the insured tocontinue the policy.

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(a) Temporary Assurance

1 Mx –Mx+n Px:n¯| = ---------------

Nx – Nx+n

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(b) Pure Endowment Assurance

1 Dx+n

Px:n¯| = -----------------Nx – Nx+n

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• Small addition is made to the Net Premiumcalculated on the basis of a selected table ofmortality and this is called the loading for adversemortality experience or simply “MortalityLoading.”

• The balance premium after outgoes of variouskinds would be invested and earn some interest.

• The effect of interest rates on insurance premiumis also significant.

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If interest incomes are not taken into accountthen the premium would be very high,probably an unattractive proposition for theproposers/ policyholders.

To avoid fluctuations in the interest earningsthe insurers assume slightly lower rate ofinterest but normally the experience would bea better one than what was assumed and thisdifference is called the "Interest loading".

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LOADING FOR EXPENSES

The expenses of the Insurer being incurred in

connection with soliciting, procuring andadministering the business have to be met out of thepremiums paid by the policyholders.

The expenses known as „Initial Expenses would be

falling under two broad categories:Expenses related directly to First YearPremium -- income expressed as percentageof premium for the first yearExpenses which can be related to New Sum

Assured -- expressed as fixed addition topremium for every unit sum assured

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The expenses during subsequent years would be muchlower when compared to those in the first year and as

such all expenses of first year can not be added to thepremium for first year alone.

A major part of the initial expenses has to be distributedthroughout the duration of the contract.

Subsequent year s expenses-- part of this is due topayment of renewal commission to the agents andtherefore expressed as a percentage of premiums.

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Other expenses incurred in administering thepolicies and servicing the policyholders arerelated to number of policies and expressed as aconstant addition to the premium per unit sumassured.

When adjustments for expenses andmodifications for contingencies as well asfluctuations are made then the resultant premiumwould be the Office Premium

Value of Office Premium = Value of Benefits+

Value of Expenses

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LOADING FOR BONUS Premiums are arrived at on the basis of mortalitiesexperienced in the past; however the actual experienceneed not be always adverse; rather the experiencenormally will be more favorable to the Insurer.

To encourage the policyholders as well as to motivatethe prospects, the Insurers introduced the concept ofsharing the Surplus with the policyholders.

Still the Insurers load the premiums on account of thebonus and such policies with definite provision forparticipation in the Surplus are known as Participating or With- Profit policies.

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Whole Life Assurance Without Profit

P1 = SAx + [I 2 – K 2 +C ] S+ K 2. Säx

äx – [I1-K 1 ]-K 1äx Endowment Assurance Without Profit

P1 = Sax :n + [I 2 – K 2 +C ] S+ K 2. Säx :n

äx :n – [I1-K 1 ]-K 1äx :n Whole Life Assurance With Profit

P1 = SAx + Sb (IA) x +[I 2 – K 2 ] S+ K 2. Säx äx – [I1-K 1 ]-K 1äx

Endowment Assurance With ProfitP1 = Sax : n + Sb (IA)x :n +[I 2 – K 2 ] S+ K 2. Säx :n

äx :n – [I1-K 1 ]-K 1äx :n

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It is assumed that payment of the sum

assured on death of the life assured ismade at the end of the year of death.

In practice the sum assured is paidimmediately on proof of death and title ofthe claimant.

Some modifications are done on theexpressions to allow for this condition.

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Age is calculated in three ways and insurersadopt any one among them for differentproducts.They are

Age last birthdayAge next birthday

Age nearer birthday

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Six steps involved in premium calculationsare:

1. Tabular premium2. Adjustments for Mode of Payment3. Sum assured rebates4. Additions like health extra, occupation

extra, and for rider benefits.5. (Modified premium * sum assured) / 10006. Divide by frequency of payments.

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ValuationIt is necessary to have periodicinvestigation into the financial positionof the insurer.

This process of investigation is called

“Valuation”. The principle feature of which isvaluation of Assets and Liabilities.

However, method chosen for valuationof Assets should be consistent withmethod for valuation of Liabilities.

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The Valuation of Liabilities of an Insurer

is the process of arriving at the policyvalues of various contracts in existenceon the date of Valuation.There are Two methods of Valuation:Prospective and Retrospective.If the values assumed are identical, boththe methods would give the same result.Normally, the Prospective method ofvaluation is adopted.

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Policy value in respect of each policy iscalculated as per the method of valuationadopted by the insurer.

Valuation liability is the total of suchpolicy values in respect of contracts inforce on the date of Valuation.

It is necessary to provide for specialreserves and adjustments as under toarrive at the final valuation liability.

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(1) Allowance for actual incidence in premiumincome.

(2) Adjustment for outstanding premiums.(3) Reserve for early payment of claims.(4) Reserve for revival of lapsed policies(5) Provision for expenses in respect of fullypaid up policies.(6) Provision for expenses in respect of reduced

paid up policies.(7) Reserve for elimination of negative values.(8) Exchange fluctuation reserve.

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SURPLUSValuations are rarely done on the basisassumed for calculation of Premiums.Even if the experience assumed inpremium calculations has beenreproduced, it is not usual to adopt thesame rates of Mortality, Interest andExpenses in the valuation of net liability.A proper valuation has to take into accountnot only the actual mortality experiencebut also future trends of mortality.

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Even with the past experience, it is notpossible to predict future trends with anydegree of certainty.Future experience may turn out to beworse.Losses may easily arise.To prevent this contingency, a life table isused based on heavier rates of mortalitythan those actually being experienced.This method provides a safety margin foradverse fluctuations.

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Rate of interest based on conservativeestimate, is adopted for valuation.Rate of interest depends upon many factors likesupply and demand, fiscal and taxation policy ofthe Government, differences in prices between

home and world market .New investments in future may not bemade at favorable rates.

It is customary to use in the valuation arate of interest which is lower than nowbeing earned.

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Valuation interest rate should be kept low

as a precautionary measure.Increased interest earnings would enablethe insurer to declare higher rates of bonusin future.The excess of life fund over valuationliability arrived at by using bases differentfrom those employed in premiumcalculations is termed surplus as it is notprofit in real sense of the term.

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Sources Of SurplusAnything received in excess of the

valuation assumptions is surplus.Surplus from mortality arises whenexperienced rates of mortality are lowerthan those assumed in the valuation.Surplus from interest arises if actualinterest earned is more than that assumedin the last valuation.Surplus from favorable expense positionarises because of the lower expensesincurred than those provided for in thevaluation.

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Reversionary Bonus System

Bonus is declared as a percentage of SumAssured and is payable along with the sumassured.Advantages of this method are:

1. Surplus fund still remains with the insurerand can earn further interest.2. Policy holders will have interest to maintain

their policies.3. The system is easily understood by the

policyholders.

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Uniform Compound ReversionaryBonus System

Bonus addition each year is of anincreasing nature.It is declared as a percentage of sumassured and bonuses so far declared.

Bonus is in effect not on sum assured onlybut on sum assured plus bonuses so fardeclared.This gives better incentives to policyholders and encourages them to continuetheir contracts.

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Bonus in reduction of premiums

Surplus of a valuation is utilized inreduction of premiums receivable.A stage may arise when there will be nobalance premiums to be reduced further.The disadvantage is that surplus isdistributed in a way on cash basis and thusthe profit earning capacity of the insurergets gradually diminished by loss ofpremium income, and consequentdepletion of funds.

TONTINE BONUS

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TONTINE BONUSMost popular in America.Distributed after a certain period to thesurvivors then existing.This is achieved by excluding the first fewyears, say, five years, for every policy from

participation in profits.No bonus if claim arises before the periodprescribed.Policies resulting into claims at earlierdurations do not receive a portion of surplusto which they have contributed and hence theyare penalised.

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INTERIM BONUSBonuses are declared on the basis of theresults of valuation for all policies in forceas on the date of valuation.Before next valuation is finalised some

policies become claims either by death orby maturity on expiration of the term.These policies will not get any regularbonus as they are not in existence at thetime of Valuation exercise.An interim bonus is allowed at the samerate as declared in the last valuation.

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GUARANTEED BONUS

It is not in effect a bonus at all.It is only a guaranteed addition to SumAssured for every year.The policy should be in force.A policy with this benefit is having noright to share in the profits or surplus ofthe Actuarial valuations.They are in effect without profit policies.

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The rate of premium would be arrived at onthe basis of the following:Degree of hazard.Classifications of the object of insurance.Past loss experience.

Degree of Hazard

Probability of loss occurrence and also theseverity of loss.

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Classifications of the object of insurance• Categorize them in broad groups

e.g. Motor vehicles into private cars,commercial vehicles, motorcycles andscooters.

• Within the broad groups, then further sub-

division e.g. godowns according to thegoods stored i.e. non-hazardous, hazardousand extra-hazardous.

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Past loss experience

Giving a mathematical value to the risks.L ----- X 100 V

L= sum total of the losses.

V= total values of the property/object ofinsurance cover.

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Final rate of premium will be arrived at aftermaking adjustments for the following:Provision for expenses.

Commission to intermediaries.Provision for loss expenses e.g: survey fees.Loading for contingencies.Margins for fluctuations.Margins for profits.

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