How To Calculate Life Expectancy
Life expectancy is something that often comes up when talking about life insurance. In this article we’ll cover what it actually means and show an easy way to calculate it. Life expectancy is an expected value, which is more commonly known as the mean or average. Since it’s an average, you use mortality rates (probabilities of dying) to calculate it.
This is very straightforward, and conceptually no more difficult than calculating something like the expected winnings in a game with a 60% chance of winning $100 and a 40% chance of losing $50. You can multiply each outcome by its probability of occurring and then add them up (.6*$100 – .4*$50 = $40). For calculating life expectancy, the possible outcomes are the ages at which death can occur, so there are going to be a lot more than two possibilities. This does not make the calculation any more difficult, just longer.
For this article, we’ll use an example to go over various concepts and calculations. The table below shows mortality rates for a 45 year old man (these are hypothetical) at different ages. They are conditional probabilities, which means that they represent the likelihood of dying given that the individual actually survives long enough to make it to the listed age.
The “q” in the table is the standard symbol for a mortality rate. So, for example, at age 50 there is a .142% chance of dying before age 51, assuming that you’ve made it to age 50 (this is the “conditional” part). If there were 100,000 people alive at age 50, you’d expect that 142 would die within the year. This table goes to age 120, after which everyone is assumed to have died.
In order to calculate his life expectancy, we need the actual probability of dying at each age for a man who is 45 today and not the conditional probability, which is what we have above. To get those, we multiply the probability of living to a given age by the conditional probability of dying and end up with the table below.
To get the value for age 47, the 45 year old must survive to that age. This is equal to (1 – .00068)*(1 – .00082) = .9985005576. These probabilities came from the mortality table above. Given that the man lives to age 47, the mortality rate is .00096, which you multiply by the probability of living to age 47. After rounding, this gives .00096, which matches the table below. The calculations are similar for other ages, but tedious to do by hand.
One thing to notice is that these numbers add up to 1, because there is a 100% chance of death at some point. Now all that needs to be done is to multiply the age by the probability of dying at that age (add 1 to the age in the table since we’ll assume people die at the end of the year). This is the same concept as the game example above — we are just multiplying each outcome by its probability.
When you add up the numbers in this table, you get 83.71, which is the life expectancy for this example. Other than modifications you could make (e.g. allowing for deaths to occur throughout the year and the pattern at which they occur), it is a very straightforward process to calculate life expectancy. People often make insurance decisions using their life expectancy (such as the length of the term for term insurance or the guarantee duration for guaranteed universal life type products), frequently with some incorrect assumptions.
Many people assume that there is a 50% chance of living to your life expectancy. In the example in this article this is roughly the case (there is a 53.56% chance of living to age 84), but in general this is not necessarily the case. Others think that it is the age at which they are most likely to die. You should not be surprised to die at an age other than your life expectancy, as this is much more likely. For a large group of individuals, the life expectancy will be the “average” age at death, but for an individual the variance is greater.
In this example, the life expectancy age is not even the age at which you are most likely to die (you can see this in the second table, where there are multiple ages that have a higher likelihood of dying than the life expectancy age). Your life expectancy also changes as you age. Each year that you survive causes your life expectancy go up a little bit.
Of course, life expectancy varies by gender, age, health, etc. and you probably don’t want to calculate it yourself, so here is a handy calculator to help you out. (This links to an external site that is run by the Society of Actuaries and American Academy of Actuaries).