1. A simple life annuity
With your knowledge of cash flow valuation and life tables, you are now ready to tackle a first example of a life contingent financial product: the life annuity!
2. The life annuity
In chapter 1 you took up the role of an actuary. You were helping a superhero, Mr Incredible, with some calculations related to his car loan.
3. The life annuity
In a loan or mortgage, the amount that is borrowed should be paid back to the bank over a specific time window. This is called an annuity certain, because the borrowed amount, plus interest, should be returned to the bank under all circumstances.
4. The life annuity
What is the difference between such a product and a pension? In a pension, payments continue as long as the retiree is alive. The payments are called life contingent, and the product is called a life annuity.
5. Annuity vs. life annuity: mind the difference!
Here's a picture of the series of payments in an annuity certain. The cash flow vector has a fixed and known duration, and is guaranteed, as in the example of the loan for Mr Incredible's car. However, in a life annuity payments depend on the survival of the recipient. In the example on the timeline, the last payment is made at time 3 because the annuitant dies in the 4th year. Hence, the number of payments is unknown upfront and thus, stochastic.
6. Pure endowment
To determine the value or price of a life annuity you will start with the most simple example of this product, that is the pure endowment. The pure endowment is sold to an x-year-old at time 0 and will pay the annuitant 1 EUR if he is alive at time k. And this is the only benefit paid out by the product.
7. EPV of pure endowment
Your job as an actuary is to calculate the value of this pure endowment at time zero. This value is called the Expected Present Value where the term 'Expected' refers to 'expected value' and thus reflects the stochastic nature of this product. Indeed, the 1 EUR is only paid at time k if the annuitant is alive at that time. The expected present value of the pure endowment sold to an x-year-old with maturity k is the benefit, 1 EUR, multiplied with the financial discount factor, v(k), multiplied with the probability to survive up to maturity, that is time k, and this probability is kpx. The actuarial notation for the expected present value of this product is kEx.
8. Annuity vs. life annuity: mind the difference!
Do mind the difference between the present value of an annuity certain and a pure endowment. You first consider an example with a guaranteed payment of 1 EUR at time 5. The interest rate i is 3% and you will use (1+i) to the power (-5) as the discount factor. In this example the present value of the annuity certain is 0-point-86 EUR.
9. Annuity vs. life annuity: mind the difference!
For the pure endowment you also need the age x of the policyholder, 65 in our example, and the probability that he survives the next 5 years. Chapter 2 taught you how to calculate this survival probability as a product of one-year survival probabilities, running from p65 to p69. Multiplying with the discount factor then gives an expected present value of 0-dot-789 EUR. This should of course be smaller than the present value of the first product, because of the probability that a 65-year-old dies before age 70. This probability is smaller than 1. In case of death of the annuitant, the benefit should not be paid. That's reflected in the lower price.
10. Let's practice!
Now it's your turn to put this into practice.