1. The whole, temporary and deferred life insurance
We now explain the working principles behind a wide range of life insurance products.
2. A series of one-year contracts
Here's the picture again of the basic one-year life insurance product. What if the death benefit is bk instead of 1 EUR? And what if you consider a series of one-year contracts instead of just one?
3. General setting
A general life insurance contract on (x) uses a death benefit vector. The benefit b0 is paid at the end of the first year in case of death of the policyholder during the first year. b1 is paid at the end of the second year if the insured dies during the second year, and so on.
You should consider this contract as a series of one-year contracts. Each of these pay bk EUR at the end of year (k+1) if the policyholder dies between time k and k+1. Hence, their Expected Present Value follows from the reasoning in the previous video. All together, the general contract is valued as the sum of the expected present values of these one-year contracts.
4. Whole life insurance
In a whole life insurance product the death benefit is payable lifelong. On the timeline you see the basic ingredients to value this product. These are the benefits insured, the discount factors and the deferred death probabilities. With a constant benefit of 1 EUR and constant interest, the Expected Present Value of this product is denoted with Ax. x refers to the age of the policyholder at contract issue.
5. Temporary life insurance
With a temporary insurance the period of insurance is limited in time. Only in case of death during this period of, say, n years a death benefit will be paid at the end of the year of death. Again, the figure shows the building blocks to value this product. The applicable international actuarial notation is valid when the benefit is constant and equal to 1 EUR, and the interest rate is constant as well. The symbol 1 on top of the x refers to the fact that a death benefit is only payable if x dies before the end of the insured period of n years.
6. Deferred whole life insurance
The last example puts focus on a deferred whole life insurance product. Here no benefit is paid if the policyholder dies during the first u years. From time u on, a death benefit is payable lifelong, at the end of the year of death of the policyholder. Again, you see the international actuarial notation at the bottom of the slide.
7. Life insurances in R
Let's do these calculations in R! You first focus on a whole life insurance product issued to a 35-year-old. The given interest rate is 3%.
You calculate the deferred mortality probabilities in the vector kqx. These are of the form kpx multiplied with qx+k. The k-year survival probabilities kpx start from 1, followed by cumprod() applied to the vector of one-year survival probabilities px. You make sure the vector of k-year survival probabilities has the same length as the vector of mortality rates starting from q35. Then you multiply both vectors elementwise. The discount factors start from v(1) and have the same length as the vector kqx. The EPV follows as the sum of the benefits vector multiplied with the discount_factors and the kqx.
Next you focus on a deferred whole life insurance product issued to a 35-year-old. Only the benefits vector is adjusted. This vector now has a benefit of 0 during the first 20 years, and then continues with a death benefit of 1 EUR. The EPV is obviously lower than the corresponding product without deferment.
8. Let's practice!
The machinery behind life insurance products is at your fingertips. Your turn!