On premium payments and retirement plans

1. On premium payments and retirement plans

You will now find out why life annuity calculations are relevant for both premium payments as well as retirement plans.

2. Paying premiums

When insurance products are bought with multiple premiums instead of a single premium, you need to calculate the value of these premium payments. The premiums plus interest earned should match the benefits promised in the contract. So, you will calculate premiums from an equivalence relation between the premium vector and the benefit vector. Since the premium payments will stop upon death of the policyholder, the premiums should be valued in the same way as the benefits of a life annuity.

3. Mrs. Incredible's retirement plan

Here's your motivating example. Mrs. Incredible is 35 years old. She wants to buy a life annuity that provides 12,000 EUR annually for life, payments should start at age 65 and are paid at the start of the year. She will finance the product with annual premiums, payable for 30 years, beginning at age 35. Premiums reduce by one-half after 15 years. What is her initial premium?

4. Mrs. Incredible's retirement plan pictured

Here's a picture of Mrs. Incredible's retirement plan. Life annuity benefits are deferred and will start from age 65 on, if she is alive at that time. The yearly benefit is 12,000 EUR. Benefits are lifelong. Premium payments start from time 0 and continue until age 64. That is 30 years in total. The first 15 premiums are denoted with P and the last 15 premiums are 0-point-5P, or half of the initial premium.

5. Mrs. Incredible's retirement plan in R

Your building blocks are set up in R in exactly the same way as in the previous video. Of course, you should now use a female life table. Let's take the one for Belgium in the year 2013 and assume an interest rate of 3%. You calculate the multi-year survival probabilities of a 35-year-old, starting from 1. And you also need a vector with discount factors. This vector has the same length as the vector with the survival probabilities.

6. Mrs. Incredible's retirement plan in R

So, two elements are required to calculate Mrs. Incredible's premium: the expected present value of her benefits and the expected present value of her premiums. The equivalence between premiums and benefits will then deliver the unknown initial premium P. The benefit vector has 30 zeros, followed by 12,000, such that the benefit vector has the same length as the vector with the survival probabilities. The expected present value of the life annuity benefits is calculated as before. You calculate the sum of the elementwise product of the benefits, the discount factors and the survival probabilities. The premium payments can be described by means of a premium pattern vector called rho. Multiplied with the initial premium P this vector gives you the stream of premium payments. Thus, rho starts with 15 ones, followed by 15 one halves, followed by zeros to make sure rho has the same length as kpx. The premium payments should be valued in the same way as life annuity benefits, so the expected present value of the premium pattern vector rho is the sum of the elementwise product of rho, the discount factors and the survival probabilities.

7. Mrs. Incredible's retirement plan in R

To solve for the initial premium P you rely on the actuarial equivalence of the expected present value of P times rho and the expected present value of the benefits. Thus, P follows as the ratio of both expected present values. The ratio of the expected present value of the benefits and the expected present value of the premium pattern leads to an initial premium of 4,427 EUR. To get a lifelong retirement income of 12,000 EUR from age 65, Mrs Incredible should pay this amount 15 times, followed by 15 times half of this amount.

8. Let's practice!

Now it's your turn to calculate some premium and retirement plans!

Create Your Free Account

By continuing, you accept our Terms of Use, our Privacy Policy and that your data is stored in the USA.