1. Combined benefits
With your knowledge of life annuities and life insurances, you are now ready to tackle products that combine both types of benefits.
2. Endowment insurance
An endowment insurance product combines a death benefit cover with a savings component. In this picture, the death benefit is constant and equals 1 EUR. It is payable at the end of the year of death if death occurs between time 0 and time n. Moreover, if the insured is alive at time n, a pure endowment equal to 1 EUR will be paid. This is the savings component in the product. The building blocks required to value such a product are: the discount factors, the deferred mortality probabilities and the survival probability npx for the pure endowment. The international actuarial notation uses x, the age of the policyholder at policy issue, as well as n, the duration of the contract. This notation is valid with constant interest and a death benefit as well as pure endowment equal to 1.
3. Sending baby Incredible to college
Let's help Mrs. Incredible by designing a suitable endowment product for her.
To send baby Incredible to college she wants to save 75,000 EUR by the time he turns 18.
Given her dangerous lifestyle as a superhero, she also wants to cover her life. This will be of great help if something untoward happens to her, the family's bread winner.
The sum insured is 50,000 euro.
4. Sending baby Incredible to college pictured
This is an example of an endowment insurance with death benefits equal to 50,000 EUR for death between time 0 and time 18. A pure endowment worth 75,000 EUR will be paid at time 18 if Mrs. Incredible is alive then. Note that she is now 35 years old and she will turn 53 at time 18. Mrs. Incredible decides to buy the product with 18 equal premiums denoted P. These premiums are payable at the beginning of the year, from time 0 until time 17.
5. Sending baby Incredible to college in R
You assume a 35-year old female, living in Belgium in the year 2013, and an interest rate of 3%.
Let's start with the expected present value of the death benefits. You need the deferred mortality probabilities, as calculated in the previous video. These run from q35, over the 1- until the 17-year deferred mortality probability.
Death benefits are valued with discount factors that range from v(1) to v(18). The death benefit vector is 18 times 50,000 EUR.
The expected present value is then the sum of the elementwise product of the benefits, the discount factors and the kqx. That is 870 EUR in this example.
6. Sending baby Incredible to college in R
The pure endowment is payable at time 18 if the 35-year-old survives to age 53. Using this survival probability, a benefit of 75,000 EUR and the factor to discount from time 18 to time 0, results in an expected present value of 42,975 EUR.
To value the premiums you need the survival probabilities of a 35-year-old, stored in the vector kpx. The first entry of this vector is 1, since the first premium is paid at time 0, and the last entry is 17p35. You combine the kpx with discount factors v(0) to v(17) and a constant premium pattern vector rho. The sum of the elementwise product of the pattern vector rho, the discount factors and the kpx is 14.
7. Sending baby Incredible to college in R
You will now set up an equivalence relation between the expected present value of the premiums and the expected present value of the combined benefits.
The constant premium P then follows as the ratio of the expected present value of the benefits and the expected present value of the premium pattern vector rho.
A level premium of 3118 EUR results.
8. Let's practice!
Time to put this into practice.