## Introduction

our approach focuses on the ‘art’ rather than the ‘science’ of insurance where we emphasize principles, concepts, and intuition rather than mathematical proofs of complex theorems. However, achieving an intuitive understanding of life insurance and life annuities depends on mastering two key principles of finance and probability theory.The first principle is a statement concerning the process of discounting future cash flows to the present time. For example, in life insurance, the policyholder typically pays regular premiums over the term of the contract starting from the date of policy issue. The issuer pays a benefit contingent on the death or survival of the insured at some point in the future. These two streams of cash flows are not synchronous. They can only be compared at a point in time and discounting cash flows to the present time is crucial.

The second principle concerns a challenging issue faced by life insurance actuaries and may be stated as a question: for an individual of age x years, what is his/her expected remaining lifetime? The answer to this question helps to determine appropriate premiums for life insurance and life annuities.

For life insurance, the policyholder pays the insurer regular premiums and receives a benefit if the insured dies within the specified term. For life annuities, payments are made in reverse whereby the individual receives regular payments from the insurer in exchange for a single premium paid to the insurance company at the date of policy issue. The payments cease upon the death of the individual.

To facilitate the presentation of the issues in this chapter, we define the parties to a life insurance contract.

A policyholder is an individual who has a contractual obligation to make premium payments to the insurer. The insured is the individual who is the source of the mortality risk that the insurer bears. In other words, the expected future lifetime of the insured determines the premium payments the policyholder makes to the insurer.

For simplicity and by common practice, we assume that the policyholder and the insured are the same individuals. That is the policyholder purchases life insurance and makes premium payments based on his/her own lifetime.

Finally, the beneficiary receives the benefit payment (also called the sum insured) if the insured event occurs. In the case of death, the beneficiary may be the policyholder’s estate.

The essential features of conventional life insurance contracts are now described. They are the building blocks on which other life insurance contracts are developed.

__1-Conventional Life Insurance__For life insurance contracts, the sum insured (also called benefit payment) is typically declared at the time the policy is issued. Therefore, uncertainty arises.

from the timing of the death of the insured. In other words, the future lifetime of the insured is a key random variable for life insurance contracts.

Periodic premium payments made by the insured are also a source of uncertainty for the insurer since the policyholder may face cash flow problems arising from, for example, becoming unemployed or disabled. As a result, the policyholder may be unable to make future premium payments. Obviously, this source of uncertainty does not exist for single premium policies, since full payment is made to the insurer at the time the policy is issued.

We now consider two fundamental life insurance contracts from which other life insurance contracts are created. The first is a ‘pure endowment insurance’ contract that is based on the survival of the individual over a term; that is, for a fixed and finite number of years.

**2-Term Life Insurance Contract****An n-year term life insurance contract is based on the death of the insured. This insurance pays a sum insured or benefits payment equal to 1 if the insured dies within n years.**

We now consider the case of a life insurance contract without a fixed term so that the insurance expires upon the eventual death of the insured. Hence, from the insurer’s perspective, the occurrence of death of the insured is certain and only the timing of death is random. This is the main characteristic of whole life insurance.

**3-Equity-Linked Life Insurance**Equity-linked life insurance is called unit-linked insurance in Europe and most of Asia, segregated funds in Canada and variable annuities in the USA.

Equity-linked life insurance (or more accurately called ‘investment-linked life insurance’) is a contract between a policyholder and an insurance company where the policyholder pays either a single premium or regular premiums until maturity of the contract or upon the death of the policyholder, whichever comes first.

The premiums received by the insurance company are invested in a portfolio of assets (e.g., equity, bonds, real estate) or in units of a mutual fund. The insurer makes a payment based on the evolution of the investment over the time period [0, n] where n is the maturity of the equity-linked life insurance contract. If the policyholder is alive at time n, then we have the features of pure endowment insurance. If the policyholder dies before n, then we have a term life insurance. This means that equity-linked insurance is similar to an endowment life insurance.

shows that if the insured dies at time t before the maturity date (n), then the benefit payment is typically the higher of two values: the sum insured (SI) declared at the time 0 and the (accumulated) investment value (IV) at time t.

If the insured survives the maturity of the contract (n), then the benefit payment is typically the accumulated value (IV) of the investment portfolio.

Since the accumulated investment value is not guaranteed, the policyholder bears the financial risk of the investment portfolio fully and solely. This potential for the significant financial risk borne by the policyholder can lower the demand for this insurance product and hence have a negative impact on the fee income of the insurer.

For this reason and to create a higher demand for this insurance product, the insurer may provide guarantees and options to the policyholder that serves to transfer part or all of the financial risk from the policyholder to the insurance company. This is a noteworthy point that bears emphasis.

__4-Guarantees and Options__Guarantees and options attached as riders to equity-linked insurance are commonly referred to as GMxBs meaning guaranteed minimum benefits of type x. For example, GMDB refers to the guaranteed minimum death benefit and GMWB is an acronym for guaranteed minimum withdrawal benefit. These are two examples of numerous possibilities of guarantees and options.

These guarantees are riders associated with equity-linked insurance contracts and come under a common name: variable annuities (VA). For example, SPVA with a rider GMWB means a single premium variable annuity with a guaranteed minimum withdrawal benefit. This is essentially an equity-linked insurance product with a single premium and provides a guaranteed minimum withdrawal benefit.

**5 -Annuities Certain**Formally, a certain annuity is a contract between an individual and a financial institution (e.g., bank) where the individual receives payments from the financial institution that are fixed over equally-spaced intervals (e.g., months) over a known finite period of time. This annuity is called annuity certain in actuarial terminology. In exchange, the institution receives a single premium from the individual at the date the contract is issued. The individual is called an annuitant.

Simply put, a certain annuity is a stream of payments paid to the annuitant for a finite term. The present value of the payments made by the financial institution is the single premium paid by the individual.

Under the equivalence principle, the premium is set such that at the start of the contract the expected value of the future loss is zero. This implies that the expected present value of premiums paid to the insurer is equal to the expected present value of benefits paid by the insurer to the insured.

Alternatively, at the date of policy issue, the actuarial present value of current and future premiums is equal to the actuarial present value of future benefits. The premium so obtained is called the pure premium. Pure premiums are premiums determined by the equivalence principle, ignoring expenses or profit margins that the insurer may add on.

The key point is that the premium (set according to the equivalence principle) will be sufficient to cover the average loss per contract at the date of policy issue.

__6- Premiums for Life Insurance__, the two fundamental life insurance contracts are pure endowment insurance and term life insurance. These life insurance contracts are the building blocks on which others are created. We begin with the procedure for calculating the premiums for these contracts and then follow with special cases of whole life and endowment insurance.

The benefit payment is assumed to made at the end of the year of death. The advantage is that we can utilize the information for mortality tables using a life table. As in Chapter 3, we will use the ILT model with an assumed interest rate of 6%.

In addition, annual premiums are discrete and paid to the insurer at the beginning of each year and so have similarities to life annuities due.

__7-Pure Endowment Insurance__Pure endowment insurance pays a fixed benefit of B=1 at the end of the nth year if the policyholder survives at least n years.

For reference, time 0 is the date the insurance contract is issued. The most common premium principle in life insurance is the equivalence principle which states that the single premium at time 0 equals the expected present value of the benefit payment.

__8- Term Life Insurance Contracts__Case of a Single Premium

We begin with an intuitive discussion to discover the main issues involved in obtaining a single premium for a term life insurance. Here is an example that facilitates our discussion.

Suppose a policyholder currently of age 70 years purchases a 5-year term life insurance. What is the single premium for this insurance policy?

We present an intuitive discovery process that will lead to a simple formula for the single premium.

For a 5-year term life insurance, there are five future outcomes stated as follows:

__The policyholder dies during the first year of the policy. Payment of a sum insured of 1 is paid by the insurer to the beneficiary at the end of the first year.__

**A:**The present value of the sum insured is

__The policyholder survives the first year and dies during the second year. Payment of a sum insured of 1 is paid by the insurer to the beneficiary at the end of the second year. The present value of the sum insured is.__

**B:**__The policyholder survives the first two years and dies during the third year. Payment of a sum insured of 1 is paid by the insurer to the beneficiary at the end of the third year. The present value of the sum insured is.__

**C:**__The policyholder survives the first three years and dies during the fourth year. Payment of a sum insured of 1 is paid by the insurer to the beneficiary at the end of the fourth year. The present value of the sum insured is.__

**D:**__: The policyholder survives the first four years and dies during the fifth year. Payment of a sum insured of 1 is paid by the insurer to the beneficiary at the end of the fifth year. The present value of the sum insured is.__

**E**These five possible deferred mortality outcomes are mutually exclusive. Only one of these five outcomes can occur. Importantly, they are all based on the application of the concept of deferred mortality.

Since the five possible outcomes are mutually exclusive, the single premium for this 5-year term life insurance is obtained by adding the values of the five cells in the last column in the table.