Life Expectancy 101

17Nov/0919

Life Expectancy 101

So, what's the advantage of having an actuary around if they won't do some actuarying for you?

There are all sorts of flavors of actuary, and my specialty is annuities [don't ask me about auto or home insurance - that's not my racket]. So one of the big things we have to learn about are survival probabilities and life expectancies. Life expectancy is a number that's been thrown around a lot in the health care/health insurance debate [We suck! The U.S. life expectancy is lower than other nations!], Social Security debate, public pensions debate, demographics discussion in general.

If you recall, the issue of John McCain's life expectancy came up during last year's elections, and I wrote about it on my own blog, coming up with a survival probability for McCain through one term: 80 - 87.7%; two terms: 60.5-74.5%. [To see actuaries discussing this in our own forum, check out this link.]

So I'm going to start at the beginning, keeping in mind that I'm simplifying the calculations greatly so as to focus on the important features. In this post, I'm just going to go over some basic life expectancy stuff.

Life expectancy is the expected value of lifetime left [duh], but this does not mean the age you're most likely to die at, and it doesn't need to be calculated from birth. You calculate this by multiplying the probability of having a specific lifetime left times the amount of that lifetime, then add those all up - that's your life expectancy, a weighted average.

To do a real simple model, let's say you can die either at ages 40, 60, 80. Let's look at life expectancies for different mortality patterns.

Lifetime Probabilities
Age	Group A	Group B	Group C
40	1/3	1/2	0
60	1/3	0	1
80	1/3	1/2	0
Life expectancy	60	60	60

You'll notice they all have the same life expectancies - 60 = 1/3 * 40 + 1/3 *60 + Â 1/3 *80 = 1/2Â * 40 + 1/2Â * 80 = 1 *Â 60. But very different mortality patterns. As with most statistics, a single number is misleading -- if you knew the standard deviation of the lifetimes, then you would know everyone dies at the same age for group C, and that group B is the most spread out.

Thing is, if you're looking at life expectancies for public policy, looking at life expectancy from birth can be misleading. Actuaries can calculate life expectancy from any age, and of course the average age at death will be going up a little for each higher age that's calculated... as all the people who made it to age 40 obviously didn't die at age 20 [except in the actuarial zombie flick I'm thinking of calling "Night of the Living Death Benefits", but that's still in spec].

So what if we wanted to know the life expectancy of a 50-year-olds in each of these groups? I can readjust the probabilities [and change the lifetimes - life expectancy is usually termed in age at death in the media, but it's really how many more years til death from whatever age]. You readjust the probabilities by taking only the possibles [living to age 60 or 80, which is extra lifetimes of 10 or 30], so you'll need to divide by the probability you survive to age 50 [which differs in each of these cases].

The results:

Lifetime Probabilities from Age 50
Time til death	Group A	Group B	Group C
10	1/2	0	1
30	1/2	1	0
Life expectancy	20	30	10
Expected age at death	70	80	60

So the different patterns become more clear once you look at life expectancy from a higher age.

This is some basic math setting up for my next few posts that look at the impact of having certain demographics in a population and how that will affect overall calculated life expectancy, cancer survivorship, Social Security and life expectancy, pensions, etc.

Meep

Meep is a member of the Irish Catholic mafia, having a suspiciously high number of green-eyed, red-haired friends. While she doesn’t have red hair herself [except when she goes into the sun (rare for any vampire)], she does have green eyes. She’s a raving Papist and is a life actuary on the side [i.e., she counts dead people]. An amateur pain-in-the-ass [willing to go pro!], she likes covering retirement, mortality, math, and education issues.

Filed under: Too Lazy to Categorize Leave a comment

Comments (19) Trackbacks (2) ( subscribe to comments on this post )

Enoch_Root
November 17th, 2009 - 07:01

I’ll take a double half-caf, latte-mochachino, with whipped cream, then. Actually, make it a decaf – vente. Ahhh, f88k it, gimme a Guinness with a pack of American Spirits.

Like or Dislike: 0 0
( REPLY )
Dan Collins
November 17th, 2009 - 07:07

LOL, Enoch.

Thanks, Meep. Don’t be afraid to go all mathy on us, just so long as you take it slowly.

Like or Dislike: 0 0
( REPLY )
Bob Reed
November 17th, 2009 - 07:10

Do the groups actually represent anythig? Or are they just artifices used for illustration of the different distributions?

My guess is the latter, but I’m curious all the same…

Like or Dislike: 0 0
( REPLY )
- Enoch_Root
  November 17th, 2009 - 07:16
  
  also – are there any artifices in there with really nice gams?
  
  Like or Dislike: 0 0
  ( REPLY )
- Meep
  November 17th, 2009 - 08:43
  
  Yes, these are artificial numbers. I’m going to have to start out with extremely artificial examples before I make it to real stats.
  
  I’m just trying to show how various numbers may be misleading in policy debates. I’m going to try to take this small step by small step.
  
  Like or Dislike: 0 0
  ( REPLY )
Enoch_Root
November 17th, 2009 - 07:15

I would like to know, Meep, if there is a way to statistically prove/disprove/demonstrate a “Judas Ratio” – it is the contention that in every human creature – and therefore every group comprised of human creatures – there is approx 8.5% corruption. Something I have been toying with for years, but wonder if there is any “there” there – I know it is not maybe your forte, and I dont even know how to approach using “lies, damn lies, and statistics” toward the end of exploring it.

Like or Dislike: 0 0
( REPLY )
- Meep
  November 17th, 2009 - 08:42
  
  Hmmmmm….. that is beyond the scope of the data I have access to.
  
  Like or Dislike: 0 0
  ( REPLY )
  - Enoch_Root
    November 17th, 2009 - 10:22
    
    could you infer Data?
    
    Like or Dislike: 0 0
    ( REPLY )
    - Meep
      November 17th, 2009 - 10:48
      
      To make a linkage to financial modeling, I think one will see that you will get different amounts of corruption displayed depending on the system.
      
      In finance, often you can infer various correlations between values… until catastrophe, at which point all correlations go to 1 or -1. I think human corruption works similarly. You can have some low level of inherent corruption bubbling around, under tight rein, and then you can have Chicago.
      
      Like or Dislike: 0 0
Mr. Bingley
November 17th, 2009 - 07:41

Enoch, it seems to me the corruption rate is 100%. Peter’s denial of Christ was every bit as corrupt and fallen as Judas’.

Like or Dislike: 0 0
( REPLY )
- Meep
  November 17th, 2009 - 08:44
  
  What about the other apostles?
  
  [yes, I know they all scattered... except, perhaps, the "beloved" one [likely John]]
  
  Like or Dislike: 0 0
  ( REPLY )
  - Mr. Bingley
    November 17th, 2009 - 08:51
    
    exactly my point :)
    
    Like or Dislike: 0 0
    ( REPLY )
- Enoch_Root
  November 17th, 2009 - 10:21
  
  did Christ say about Peter “it would be better that he not be born”?
  
  Like or Dislike: 0 0
  ( REPLY )
  - Mr. Bingley
    November 17th, 2009 - 11:23
    
    no, he said ‘get behind me, satan’
    
    the point is that if peter, who actually walked along side the Lord, could so easily and quickly deny him when the going got tough, regardless of his later redemption and valiant struggles in His name, well, we all have corruption and sin in our core and can fall prey to it, can give in to it.
    
    Like or Dislike: 0 0
    ( REPLY )
    - Enoch_Root
      November 17th, 2009 - 14:08
      
      agreed – I am saying that on average 8% of each person may be corrupted as a matter of course. Some 1%, and at the other end of the spectrum 17% – anyway, it is some degree of 1/12th
      
      Like or Dislike: 0 0
    - Mr. Bingley
      November 17th, 2009 - 14:23
      
      I’m sad to say I’ve gone over that 17% number a few too many times, especially during college…
      
      Like or Dislike: 0 0
    - Enoch_Root
      November 17th, 2009 - 15:23
      
      me too – me too.
      
      Like or Dislike: 0 0
    - meep
      November 18th, 2009 - 16:30
      
      Okay, look, I’m as Catholic as the next POWIP blogger [well, some of us], and I had a great time discussing in RCIA whether mothers have stretch marks in heaven, but I AM NOT MIXING ACTUARYING AND CATHOLICISM!
      
      [yet]
      
      Like or Dislike: 0 0
Charles Pergiel
November 18th, 2009 - 18:43

At last! Someone who might know something about death rates!

I was looking into death rates not too long ago and I came across this page of statistics on the CIA web site.

The numbers are a little confusing. For instance:

#1 is Swaziland, with a death rate of 30.35. That translates to a life expectancy of roughly 33 years (1000/30.35). I can understand that. There is a very high likelihood of dying there.

#222 is United Arab Emirates with a rate of 2.16, which translates to a life expectancy of 463 years, which is obviously wrong.

From this simple calculation, it is unlikely that any country should have a death rate below 12 or 13, but most of the countries on this list fall into this category.

So either old people from all over the world are emigrating to Swaziland to die, or every woman of childbearing age is giving birth to quadruplets every year, or I am missing something.

Any ideas?

Like or Dislike: 0 0
( REPLY )