1. A simple life insurance
A life annuity provides an income while the annuitant is alive. The second example of a life contingent financial product studied in this course pays a benefit upon death of the insured. This is the life insurance!
2. The life insurance
In a life insurance contract, the policyholder receives protection or coverage from the insurance company in the form of a death benefit that is paid to his beneficiaries in case he dies. Typical beneficiaries are: the policyholder's children or beloved ones, his bank or his business partners. Thanks to the insured capital, the children can for instance finance their studies in case of death of a parent, or coworkers may continue the business in case of death of one of their business partners. When the insured is paying off a loan or mortgage, the life insurance will pay out the outstanding balance on the loan to the bank in order to settle the mortgage.
3. A simple life insurance
Let's start with the most simple type of contract. Consider a life insurance contract sold to an x-year-old at time 0. This simple contract will pay 1 EUR at the end of the year of death of the policyholder if death occurs between time k and k+1. That is: if death occurs at age x+k. This scenario happens if the policyholder first survives to time k and then dies within one year.
4. A simple life insurance
In the previous chapter you learned that the expected present value of a life annuity relies on both discount factors and survival probabilities. The valuation of a life insurance product works exactly the same way. First, discount factor v(k+1) is used to discount the 1 EUR at the end of year k+1 to the present moment. Second, you require the probability that x will survive for k whole years, but will die before age x+k+1. That is the product of kpx and qx+k. Recall from the chapter on life tables that this is a deferred mortality probability, also denoted with k|qx. The international actuarial notation for the expected present value of this product is shown on the left-hand side of the equation.
5. A simple life insurance in R
How should you implement this reasoning in R? Your job is to value a life insurance product sold to a client who is 65 years old at time 0. The product pays 1 EUR at time 6 if the insured survives for 5 whole years, but then dies between age 70 and 71. The given interest rate is 3%.
First, you extract the one-year mortality rates qx and the corresponding one-year survival probabilities px from the life table. From these, you calculate the 5-year deferred death probability of the 65-year-old. To get the deferred probability, you calculate the 5-year survival probability as the product of the one-year survival probabilities p65 until p69. You then multiply this survival probability with the mortality rate of a 70-year-old to obtain the 5-year deferred death probability.
6. A simple life insurance in R (cont.)
Second, you calculate the factor that discounts 1 EUR paid at the end of the sixth year to the present moment. Indeed, if the policyholder survives 5 years, and then dies between age 70 and 71, the death benefit is paid at time 6. The expected present value of this simple product is then the benefit multiplied with the discount factor and the deferred mortality probability.
7. Let's practice!
This is the basic reasoning behind life insurance products. Now it's your turn to put this into practice!