1. The whole, temporary and deferred life annuity
What about more general products that pay a benefit contingent on survival? This video explains the working principles behind life annuities.
2. A series of benefits
Here's the picture of the pure endowment again. What if the benefit is ck instead of 1 EUR? What if you consider a product with a series of payments instead of just one?
3. General setting
You consider a life annuity for an x-year-old where the benefit ck is paid if the annuitant is alive at time k. To buy this product picture it as a sequence of pure endowments, each with benefit ck and corresponding present value. All together the expected present value of the life annuity is the sum of the expected present value of these pure endowments, with k running from 0 to +infinity.
4. Life annuities in R
Consider a life annuity with benefit vector given on top of the timeline. The product is sold to a 65-year-old and the interest rate is 3%. The applicable discount factors and survival probabilities are pictured below the timeline.
To value this product you specify the vector of benefits.
The discount factors are 1 plus 0-point-03 to the power minus a vector from 0 until 7.
You then define the relevant survival probabilities. These start from 1, the probability to be alive at time 0, followed by the cumulative product of the px vector from the preloaded life table to get 1p65 until 7p65. The price of the life annuity follows as the sum of the elementwise multiplication of these vectors.
5. Whole life annuity due
Let's consider specific examples of life annuities. In the whole life annuity due payments are made at the beginning of the year, starting from time 0. Moreover, the payments continue as long as the annuitant is alive. The actuarial notation for the expected present value of this product in case of a constant benefit of 1 EUR is shown here.
6. Whole life immediate annuity
In the whole life immediate annuity payments are made at the end of the year, instead of the beginning. Thus, the first benefit is paid at time 1 but only if the annuitant is alive, and so on. The actuarial notation is the same as before, but without the double dot.
7. Whole life annuities in R
Now picture a 35-year-old and value the whole life annuity due. That is the life annuity that pays at the beginning of the year. First, you calculate the probabilities to survive to time 0, time 1, time 2 and so on, as the combination of 1 and the cumprod() of the vector px starting from p35. Second, you load the discount factors which discount from time zero to zero, time 1 to zero and so on. The vector discount_factors has the same length as kpx. The benefit is constant and equals 1. The sum of the elementwise product of benefits, discount factors and survival probabilities gives the expected present value.
For the immediate annuity the reasoning is similar, but with the first benefit paid at time 1. Hence, the first survival probability and the first discount factor refer to time 1. The difference between both expected present values is exactly 1 because the annuity due has one extra term: the benefit at time zero.
8. Temporary life annuity due
With a temporary life annuity due the benefit vector runs from time 0 to time (n-1). The payments stop from time n, even if the annuitant is alive. Actuarial notation explicitly takes this duration n into account.
9. Deferred whole life annuity due
The last example is a deferred whole life annuity due where benefits only start after a period of deferment with length u. The first payment is now at time u, if the annuitant is alive at that time.
10. Let's practice!
Wow! The machinery behind life annuities is at your fingertips. Let's practice!