David J. Kupstas, FSA, EA, MSEA Chief Actuary
When people hear that I am an actuary, the first question they ask (besides, “What is THAT?”) is, “How long am I going to live?” The answer is, I have no idea. Contrary to popular belief, life expectancy rarely is used in actuarial calculations.
Still, mortality tables are a very important part of actuarial work. This article explains a little bit about what mortality tables are and how we use them.
Mortality Table Basics
In its most basic form, a mortality table consists of two columns of numbers. The first column is a listing of all ages from 0 to 120. The second column contains a q value for that age, which is the probability of a person that age dying in that year.
The death probabilities at most ages are very low. According to the 2017 IRS applicable mortality table, the probability of someone dying while age 30 is less than three-hundredths of a percent; out of 10,000 30-year-olds, three are expected die in the next year. On the other hand, the probability of someone dying at age 90 is more than 14 percent (14 out of 100 will die). If you are lucky enough to reach age 120, the table assumes you have a 100% chance of dying in the next year. Sorry!
People with different characteristics are expected to die at different rates. Separate mortality tables can be constructed for males and females, smokers and non-smokers, people in one state or country vs. another, different industries, income levels, and more. Data may be obtained from sources such as Social Security, insurance companies, actuarial consulting firms, and public pension plans. Most pension actuaries do not construct mortality tables themselves, but rather use standard tables that have been prepared by others.
Present Value with a Twist
Present value is what an amount of money in the future is worth in today’s dollars. Assuming 4.00% interest, the present value of $100.00 payable in five years is $100.00 ÷ 1.04^{5} = $82.19. Actuarial present value is present value with a twist. With an actuarial present value, not only are you discounting future payments with interest, you are also reflecting the uncertainty that a given payment will even be made. If there is only a 50% chance that the $100.00 mentioned above will be paid in five years, the actuarial present value of that payment is 50% x $100.00 ÷ 1.04^{5} = $41.10.
Let’s do an actuarial present value example using what we’ve learned about mortality tables. Because mortality means death, we are going to refrain from using an example in which a person might die. After all, young children may be reading this post. Instead, our example will use a tree – specifically, a money tree.
Suppose someone offers to sell you a money tree which will produce $1,000 at the end of each year it is alive. The tree can live for as long as 10 years. It might live 10 days, it might live four years, or it might live close to the full 10 years. How much should you pay for the money tree? To determine a fair price, you might want to find the actuarial present value of the money the tree could produce.
If you buy the money tree, assume you could have invested what you paid for the tree in an account earning 4.00% per year. And let’s say you’ve read up on money trees and know the chances of one surviving each year from now until year 10. You do an analysis like the following:
Year (t) |
q_{t} |
p_{t} |
l_{t} |
v^{t} |
Harvest |
Pres. Val. Harvest |
Act. PV Harvest |
1 |
10% |
90% |
90.0000% |
96.1538% |
$1,000.00 |
$961.54 |
$865.38 |
2 |
15% |
85% |
76.5000% |
92.4556% |
$1,000.00 |
$924.56 |
$707.29 |
3 |
20% |
80% |
61.2000% |
88.8996% |
$1,000.00 |
$889.00 |
$544.07 |
4 |
25% |
75% |
45.9000% |
85.4804% |
$1,000.00 |
$854.80 |
$392.36 |
5 |
30% |
70% |
32.1300% |
82.1927% |
$1,000.00 |
$821.93 |
$264.09 |
6 |
40% |
60% |
19.2780% |
79.0315% |
$1,000.00 |
$790.31 |
$152.36 |
7 |
40% |
60% |
11.5668% |
75.9918% |
$1,000.00 |
$759.92 |
$87.90 |
8 |
50% |
50% |
5.7834% |
73.0690% |
$1,000.00 |
$730.69 |
$42.26 |
9 |
50% |
50% |
2.8917% |
70.2587% |
$1,000.00 |
$702.59 |
$20.32 |
10 |
100% |
0% |
0.0000% |
67.5564% |
$1,000.00 |
$675.56 |
$0.00 |
Total |
$3,076.01 |
That’s a bunch of numbers and symbols. Let’s do a little explaining. Or you can just skip past the bullet points and rejoin us in the last sentence of this section.
- q_{t} is the probability that the tree will die in year t. There is a 10% chance the tree will die before it is a year old. There is a 30% chance that a tree surviving four years will die in its fifth year. There is a 100% chance a tree that makes it to nine years old will die in its 10^{th} year.
- p_{t} is the probability the tree won’t die in year t. It equals 100% minus q_{t}.
- l_{t} is the probability the tree is alive after year t. This is the product of all the p_{t}’s prior to and including year t.
- v^{t} is the fraction of a dollar someone would need to invest today to have a dollar after year t if money grows with interest at 4.00% per year. It takes around 82 cents invested at 4.00% to have a dollar after year 5. Stated another way, a dollar at year 5 is worth around 82 cents today.
- The Harvest column shows the $1,000 the money tree would pay at the end of each year.
- Pres. Val. Harvest is what each future $1,000 harvest is worth in today’s dollars assuming 4.00% interest. For each year t, it is Harvest × v^{t}.
- Act. PV Harvest for each year t is $1,000 × v^{t} × l_{t}. Here, not only are the future $1,000 harvests discounted at 4.00% interest, they are discounted further to reflect the probability that the tree won’t be alive to produce the harvest in a given year.
The sum of the actuarial present values of each year’s $1,000 that could be yielded by the tree is $3,076.01. Upon completing this analysis, you might decide that $3,076 is a fair price to pay for the money tree.
Examples That Are More Real-Life
Of course, this is not a typical problem an actuary would work on. Haven’t you been told that money doesn’t grow on trees? But there are plenty of real-life examples of when an actuary would use a mortality table to determine an actuarial present value. Life insurance is a classic example. You know that an insured person is going to die sometime, just not when. One has to figure out the amount of death benefit payable in each future year, the present value of each year’s death benefit, and the probability using a mortality table that the insured person will die in each of those years. From there, the premium can be determined.
An example near and dear to our hearts is lump sum payments from a pension plan. When a plan participant retires, he usually may choose from several different monthly pension types, including single life annuity and joint and survivor annuity. Many plans also offer lump sum settlements. In lieu of the monthly pension, a participant may choose an immediate single large payment and have no further claims to any benefits from the plan. The lump sum is an actuarial present value of the monthly pension based on interest and mortality assumptions that are prescribed by the government. Like the person thinking about buying the money tree, the pension plan retiree has to decide which is the better deal between the lump sum payment now or a monthly pension payable for however long he and/or his beneficiary live.
The IRS has written proposed regulations which, if enacted, would make significant changes starting in 2018 to the mortality tables used in lump sum payouts. We will discuss these proposed regulations in a future blog post.