Insurance Models for Joint Life and Last Survivor Benefits

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Insurance Models for Joint Life and Last Survivor Benefits

Andrejs Matvejevs and Aleksandrs Matvejevs

Riga Technical University, Latvia,

Keywords: Joint life assurance, pension benefits, surrender value, last survival assurance.

ABSTRACT: Three variants of the insurance policy for two individuals – man and woman for the calculation the net premiums according to each policy variant is considered. The net premium equation principle is used in all premium calculations. The particular qualities of the additional pension assurance is the individual form of its undertaking and the limitation of annual (monthly) pension payments. Due to this fact the biggest interest to the individual insurance shows people after age 40-45 years, when the insurance premium rates are so high that most people can't buy such policies. So, the discussed form of the joint life insurance could be proposed to the participant of the pension plan when he or she is reached the pension age and wants to buy the life insurance policy for the accumulated capital of pension.
Nowadays the family insurance contracts have a wide application when, as against individual insurance, the contract covers a minimum two persons. In theory it is quite possible to have more lives assured if insurable interest exists. These types of joint life contracts are either based on the first death or the second death. A joint life first death policy pays out on the death of the first of the two lives assured. First death term assurance and family income policies are used for family protection properties, and first death endowments are commonly used in connection with house purchase arrangements. A joint life second death policy, sometimes referred to as a joint life last survivor policy, is often used to plan for provision against inheritance tax and sometimes for investment. Where retirement provision is required, joint life and last survivor annuities have been developed to ensure that the annuity payments continue to the surviving partner after the death of their spouse. These annuities can be in advance or arrears, with or without proportion and with or without guarantee, as for single life annuities.

The formation of the proportions between premiums and benefits in the life assurance contracts of two persons are based on the general actuarial theory. The specificity here consists that the value of the net premium depends on probability of certain events, relating to cumulative life of two persons. The symbol npxy designates probability that two persons, one of age - x, and other – of y, will live simultaneously - n years, i.e. that their cumulative life will not be disturbed during n years of a moment of achievement of age x and y correspondently.

The considered event will come, if there will be two events: the person in the age of x will live up to age x + n , and person in the age y - up to age y + n. Probabilities of each of these events are npx and npy correspondently. These probabilities are independent, so n pxy = n px n py.


The insurance contract consists with a married couple - man of age x and woman of age y. According to the contract the participants of the insurance contract are obliged to pay the annual premium at a rate of P Ls during n years while both are alive.

If after the expiration period of n years both participants of the contract are alive, i.e. have reached the age of x+n and y+n accordingly, the lump sum Q Ls is paid and therefore the insurance contract is finished.

If one of the participants has died before the termination period in n years (period of payments), the insurance company undertakes to pay the life rent at a rate of R Ls to other spouse annually. There are two possibilities to start payment of benefits:

- at the end of year of death of the spouse (or in the beginning of the next year);

- upon termination of the period of payments, i.e. after the spouse is reached by

the age of x+n years (or, accordingly, y+n years).

If the death of the second spouse is occurred, the action of the insurance contract stops also and any additional payments are not made. If both participants of the insurance contract have died within one year before termination of the expiration period, then depending on conditions of the contract:

- the action of the contract is finished and any additional payments are not made;

- the premiums had paid without percents are paid to the legatee of the participants and after that the insurance contract is finished.

  • the lump sum will be return to the legatee by the insurance company in case

of death of the insured person before the benefit period.


2. Benefit

Ry Ls p. a.

in arrears





2. Benefit

Rx Ls p. a.

in arrears

1. Premium

P Ls p.a.

in advance


Lump sum кР

P P P P P P Rx Rx Rx Rx Rx

… …

Tх= k1

y х+n x




Fig. 1. The financial obligations of Joint-life annuity contract

Rxman annual benefit, if woman is died; Rx =1Ls;

Ry - woman annual benefit, if man is died; Ry =1Ls;

x – man’s age;

y - woman’s age;

n – the term of the contract.
The insured person has the right to terminate the contract at any time before to the beginning of the benefit period. In this case the surrender value will be return to the insured person by the insurance company.


There are much more various situations in the life assurance contracts of group of the persons, than in individual insurance. It is displayed in a variety of possible demographic condition for group of the persons.

Let group G, consisting from m persons of various age, is available x1, x2,... xm, where xi - the age of i person. We shall consider, that each person in group has individual number, so a vector < x1, x2,... xm > completely describes age structure of group.

After a few years some members of group can die and the number of group will decrease. Beforehand usually it is impossible to tell, who will die in the given group. In this situation it is possible only to speak about probability of distribution of death. Let i is the indicator of the demographic status, which in any moment of time accepts meaning 1 ( i =1), if the i-th member of group is still alive, and meaning 0 ( i =0), if he is not alive. Distribution of death, described by the binary cortege < x1, x2,... xm > completely defines a demographic condition all group in the given moment of time. It is natural, that in an initial moment of time all persons of group are alive and its initial condition is described by a vector 0i = < 1, 1,... 1, >. Below us will interest probabilities of various demographic condition of group. At calculation of these probabilities we shall consider carried out a condition of independence, meaning, that the demographic events for the various persons in group are in pairs independent. From this condition follows, that probability for all persons in group to live n years, designated by is equal to product of survival probabilities of each person separately, i.e.

As the usually individual rent, paid at the end of each year provided that all members of group are still alive, and, designating its current cost by a symbol , it is possible to write down

. .

Completely similarly, considering the life insurance of group for sum assured S = 1, paid at the end of a year, in which there will be the first death of which or person from group, it is possible to receive actuarial cost of such contract, i.e.

With respect to an insurance policy we define the total loss L to the insurer to be the difference between the present value of the benefits and the present value of the premium payments. This loss must be considered in the algebraic sense: an acceptable choice of the premiums must result in a range of the random variable L that includes negative as well as positive values.

A premium is called a net premium if it satisfies the equivalence principle

i.e. if the expected value of the loss is zero.

The fundamental rule to be followed in calculation of net premium in insurance practice is to specify the insurer’s loss L, and then to apply the equivalence principle. Thus, the premium P and the benefits R in formulated above model can be calculated from the condition of equality of the financial obligations between the insurer and the insured person on a moment of the contract.
Expected present value of future premiums:

Component means the payments of the annual premium Р in advance during n of years with the annual interest rate i.

- probability, that (x) died during t years;

- probability, that (x) survive t years and died during the next year;

- n-year joint life annuity, if joint life status is not change.
Expected present value of future benefits:

- n-year pure endowment joint life assurance contracts;
- n-year deferred annuity;

- n-year increase joint life assurance contract.

Net premium equation:


Developed algorithms of net premiums calculation for joint life assurance of two and more persons can be successfully applied in the life insurance companies. For this purpose, certainly, it is necessary to supply, that all accounts are automated. Therefore a database in environment MS EXCEL 2000 for the net premiums calculations of joint life assurance is created. Due to simplicity of use and presentation of results MS EXCEL is one of the most popular programme products for designing the financial reports.

The above mentioned database carries out the following functions:

  1. Calculation of the insurance tariffs of joint life insurance: net premiums

calculation for various kinds of the insurance contract.

2) Calculation of expected values of the future premiums and benefits.

3) Calculation of mortality parameters for men and women.

4) Calculation joint mortality probabilities for men and women.


  1. Neill, A. (1989). Life contingencies. The Institute of Actuaries and the Faculty of Actuaries in Scotland.

  2. Bowers, N., H.Gerber, J.Hickman, D.Jones, C.Nesbitt (1986). Actuarial Mathematics. The Society of Actuaries, Itasca, Illinois.

  3. Фалин, Г.И. (1996). Математические основы теории страхования жизни и пенсионных схем. Москва.

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