Universidad Carlos III de Madrid

Licenciatura en Ciencias Actuariales y FinancierasSurvival Models and Basic Life Contingencies

PART II. Lecture 4:Net Premium Calculations

This is a summary of formulae to calculate single net premium (expectedpresent value) and variance for assurances and annuities. Below we willintroduce the notion of yearly and monthly net premium. In the following, weassume that we have a life table containing actuarial functions: lx, dx,Dx, Ax,and ax at our disposition.

Remark 1 Remember that listed formulae correspond to contracts withpayments equal to 1. If the yearly payment realised is, in general,C (single payment for assurances or periodic payment for annuities),single net premium for such a contract has to be multiplied by C andthe variance of its present value has to be multiplied by C2.

Example 1 Single net premium of a whole life insurance paying 1 at theend of the year of death issued for a 40 years old person is

A40

If the ammount paid at the end of the year of death were C=100 thesingle net premium would be

C=100 ·A40and the variance of its present value

C=10.000 · £2A40 − (A40)2¤ .1

1 Assurances

1.1 Discrete Assurances

n-year Pure Endowment pays an amount at the end of the n − th yearafter the policy issue if the insured survives until that time.Whole Life Insurance pays an amount at the end of the year of death ofthe insured.m-year Deferred Whole Life Insurance pays an amount at the momentof death of the insured if death occurs at least m years after the policy issue.n-year Term Insurance pays an amount at the moment of death of theinsured if the insured dies within the n − year period commencing at theissue.n-year Endowment pays an amount either at the moment of death of theinsured or at the end of the n−th year after the policy issue, whatever occursfirst.

1.2 Continuous Assurances

n-year Term Insurance pays an amount at the moment of death of theinsured if the insured dies within the n − year period commencing at theissue.n-year Endowment pays an amount either at the moment of death of theinsured or at the end of the n−th year after the policy issue, whatever occursfirst.Whole Life Insurance pays an amount at the moment of death of theinsured.m-year Deferred Insurance pays an amount at the moment of death ofthe insured if death occurs at least m years after the policy issue.

2 Life Annuities

2.1 Discrete Life Annuities

Actuarial Perpetuity or a Whole Life Actuarial Annuity is a set ofyearly payments realised at the beginning (in case of annuity-due) or at theend of year (in case of annuity-immediate). Payments are realsed only if theinsured is alive.

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n-year temporary annuity-due is a stream of payments payable at thebeginning of the first n years after the issue. However, payments are realisedonly if the life assured is still alive.n-year temporary annuity-immediate is a stream of payments payableat the end of the first n years after the issue. However, payments are realisedonly if the life assured is still alive.m-year deferred n-year temporary annuity-due is a set of n future pay-ments; each amount is payable at the beginning of yearm+1,m+2, . . . ,m+n(counting the year from the date of issue). No payment is made during thefirst m years after the issue. Moreover, amounts are paid only if the personinsured is alive at the moment of payment.m-year deferred n-year temporary annuity-immediate is a set ofn future payments; each amount is payable at the end of year m + 1,m +2, . . . ,m+n (counting the year from the date of issue). No payment is madeduring the first m years after the issue. Moreover, amounts are paid only ifthe person insured is alive at the moment of payment.

2.2 Life Annuities paid m-thly

Actuarial perpetuity with m-thly payments is a stream payments ofvalue 1

mmade at the beginning (if it is due) or at the end (if it is immediate)

of each of the m sub-periods within a year. The payments are conditionedby the survival of the person insured.n-year temporary actuarial annuity-due with m-thly payments is astream of n×m payments of value 1

mmade at the beginning of each of the

m sub-periods within a year. The payments are conditioned by the survivalof the person insured.n-year temporary actuarial annuity-immediate with m-thly pay-ments is a stream of n ×m payments of value 1

mmade at the end of each

of the m sub-periods within a year. The payments are conditioned by thesurvival of the person insured.

2.3 Continuous Life Annuities

Continuous life perpetuity and n-year temporary continuous lifeannuities payable at a constant rate 1 are ordinary financial continuousannuities, however, payments are conditional on the survival of the insuredperson.

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name of contract symbol* single net premium variance of PV **

discrete assurancesn-year pure endowement nEx

Dx+n

Dx

2nEx − (nEx)

2

whole life insurance Ax from the life table 2Ax − (Ax)2

m-year deferredwhole life insurance m|Ax mEx ·Ax+m

2m|Ax −

¡m|Ax

¢2n-year term isurance A1

x:ne Ax − mEx ·Ax+m2Ax − 2

mEx · 2Ax+m

n-year endowment Ax:ne A1x:ne + nEx

2Ax:ne −¡Ax:ne

¢2continuous assurances%

whole life insurance Axiδ·Ax

i2+2i2δ

2Ax −¡iδAx

¢2m-year deferredwhole life insurance m|Ax mEx Ax+m

2m|Ax −

¡m|Ax

¢2n-year term insurance A1

x:neiδ·A1

x:nei2+2i2δ

2A1x:ne −

³iδA1x:ne

´2n-year endowment Ax:ne A1

x:ne + nEx2Ax:ne −

¡Ax:ne

¢2discrete annuitiesperpetuity-due ax from the life table d−2 [2Ax − (Ax)

2]perpetuity-im.*** ax ax − 1 d−2 [2Ax − (Ax)

2]n-year temporarylife annuity-due

ax:ne ax− nEx · ax+n d−2h2Ax:ne −

¡Ax:ne

¢2in-year temporarylife annuity-im.

ax:ne ax:n+1e − 1 d−2£2Ax:n+1e − (Ax:n+1e)2

¤m-year deferred n-yeartemporary annuity-due m|ax:ne mEx · ax+m:ne –

m-year deferred n-yeartemporary annuity-im. m|ax:ne mEx · ax+m:ne –

annuities paid m-thly%

perpetuity-due a(m)x ax − m−1

2m–

perpetuity-im. a(m)x ax +

m−12m

–n-year temporarylife annuity-due

a(m)x:ne a

(m)x − nEx · a(m)x+n –

n-year temporarylife annuity-im.

a(m)x:ne a

(m)x − nEx · a(m)x+n –

continued on the next page

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name of contract symbol* single net premium variance of PV **

k-year deferred n-yeartemporary annuity-due k|a

(m)x:ne kEx · a(m)x+k:ne –

k-year deferred n-yeartemporary annuity-im. k|a

(m)x:ne kEx · a(m)x+k:ne –

continuous annuities#

perpetuity ax δ−1 · ¡1−Ax

¢–

n-year temporary ax:ne δ−1 · ¡1−Ax:ne¢

–* symbol for single net premium (expected present value)** PV stands for Present Value*** im. stands for immediate% formuae for this type of contract are approximate# continuous annuities are paid at a constant rate 1

3 Yearly and Monthly Net Premium

Until, now we have assumed that the premium is paid as a single paymentat the moment of the issue of the contract. However, in many cases, thispremium is considerable amount of money and insured prefers to distributeits payment into equal yearly (monthly, quaterly, . . . ) payments.The most basic way to determine the size (amount) of the periodic pay-

ments is to use so-called zero expected loss principlewhich means thefolowing: On one hand we have contract bought by the insured. Its pay-ment (periodic in case of annuities or single in case of assurances) is C and isfixed. The net present value of the contract is a random variable C ·Z. Ourgoal is to determine the size of periodic payment P made by the insured.Notice that the payment of the insured person is conditional on her survival,therefore we can consider it as an actuarial life annuity. Its present value isalso random variable P ·W . We would like to set P in such a way that

E [C · Z] = E [P ·W ]

that isC ·E [Z] = P ·E [W ]

from this we have

P = C · E [Z]E [W ]

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Example 2 Imagine a 30 years old person that buys a pension plan thatguarantees C=40.000 payment every year beginning from the age 65.The insured person wants to pay the premium yearly until the age 65.On one hand, we have 35-years deferred life perpetuity-due purchasedby the insured. Its expected present value is 35|a30. On the other hand,the payments from the insured to the insurance company are a 35-year temporary life annuity-due. Its present value is a30:35e. Thus, theamount paid by the insured each year will be

P = C=40.000 · 35|a30a30:35e

.

Example 3 Let us calculate the monthly net premium P to be paid fora pension plan. The person insured is 25 years old and will pay thepremium until the retirement age of 70 years at the beginning of eachmonth. The pension of C=2.000 will be paid at the end of month.The insured gets a 45-years deferred perpetuity-due paid monthly withtotal annual paiment of 12 ·C=2.000 = C=24.000. On the other hand theinsured will pay a 45-years temporary annuity-immediate paid monthlywith total annual payment 12 ·P . Zero expected loss principle gives usthe equation (expected value of what you get is equal to the expectedvalue of what you pay)

C=24.000 ·45| a(12)25 = 12 · P · a(12)25:45e

wherefore the monthly net premium P is expressed as

P =C=24.000 ·45| a(12)25

12 · a(12)25:45e=C=2.000 ·45| a(12)25

a(12)25:45e

.

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