1. Guaranteed payments
So far you worked with the annuity certain product in chapter 1 and the life annuity in the previous videos in this chapter. This video explains how you can combine features of both products.
2. Guaranteed payments
What happens if a policyholder who buys a life annuity dies right after paying the premium? Then the money invested in the product is lost, which is a rather unpleasant outcome for his loved ones.
An interesting flexibility to incorporate in life annuities is the presence of an initial, guaranteed period where benefits are paid regardless of whether the annuitant is alive, or not.
Afterwards benefits are paid conditional on survival, as in a regular life annuity. Due to the guaranteed period this life annuity has a higher expected present value compared to a life annuity where all payments are conditional upon survival of the annuitant.
3. Mr. Incredible's prize!
Let's look at an example. Here's Mr. Incredible! He is 35 years old and recently won a special prize in the Actuarial Lottery: a life annuity of 10,000 EUR each year for life!
The first payment starts at the end of the year. And his first 10 payments are guaranteed. Can you calculate the value of his prize?
4. Mr. Incredible's prize in R
In R you first create the basic building blocks for life annuity valuation: the survival probabilities and the discount factors. Use the life table for Belgian males in the year 2013 and assume a constant interest rate of 3% for this example. You create a vector kpx with first entry equal to 1 and then the one-year, two-year and so on survival probabilities for a 35-year-old. Recall that you calculate these with the cumprod() function applied to the vector with one-year survival probabilities. The vector discount_factors should have the same length, and stores the factors to discount from time 0, time 1, time 2 and so on to the present moment.
5. Mr. Incredible’s prize pictured
Here's a picture of the cash flow in Mr. Incredible's prize. 10,000 EUR per year, with the first payment at time 1. The first 10 payments are guaranteed. From time 11 on the benefit is paid upon survival of our superhero, again at the end of the year.
You specify both vectors in R: the benefits_guaranteed and the benefits_nonguaranteed. Both vectors must have the same length as the kpx vector defined earlier on. You fill both vectors with zeros and with the yearly benefit of 10,000 EUR as seen here.
6. Mr. Incredible’s prize in R
Almost done! To get the value of this prize you first need the present value of the guaranteed benefits. Take the elementwise product of the vectors benefits_guaranteed and discount_factors and sum the elements of the resulting vector. That results in a PV of 85,302 EUR. The nonguaranteed benefits should be valued with both the discount_factors and the kpx vector. This creates an expected present value of 149,675 EUR. All together the prize is worth 234,977 EUR at time zero.
7. Let's practice!
Well done! You are now ready to apply your valuation skills in practice.