## Tuesday, July 9, 2013

### Life Expectancy Grows with Age

This post provides an illustration of a simple idea that many readers will understand, but it may still provide "Aha" moment for some readers.

The idea is that your life expectancy continues to grow as you age, so that the number of expected remaining years in your life will decline, but it declines at a slower rate than your increasing age.

This is illustrated below using median life expectancies, where there is a 50% probability of living past a certain age. This distribution is based on the Gompertz probability distribution function (which I will write more about that later time). The idea is that age 65, the person has a 50% chance of living 21.85 more years to just below age 87. The blue curve shows the remaining lifespan based on this age 65 calculation. By age 87, the number of remaining years is zero. Subsequent ages are beyond the age 65 life expectancy.

The red curve, on the other hand, shows the number of remaining years one has a 50% chance of still living, given that they've already survived to each subsequent age beyond 65. This is how things really work. Someone who is age 80 now has more than a 50% chance of living beyond age 87. They actually have a 50% chance of living to age 90. They can expect 10 more years. For someone who has lived to age 90, the remaining life expectancy is not zero. Given their age, they can still expect a 50% chance of living five more years.

This explains why it is a big mistake to use life expectancies at birth when thinking about how long one will live in their retirement. Once one has reached their retirement age, they should to live much longer than their life expectancy at birth because they were not a part of the population that died before the retirement age.