### Geeking Out: On Life Expectancy and Conditional Expectation

by meep

For this Thanksgiving, I’m going to amuse myself and explain something that confused somebody else and asked me about it.

In this TIAA paper – Income Insights:

Gender Retirement Gap there are these following items stated:

So — how can these all be true:

- Expected age at death for men and women who are now age 65 are 83 and 85.5, respectively – a difference of 2.5 years
- There generally is an age gap of 2.1 years between male/female spouses in the U.S.
- If there is a male/female couple with man age 65 and woman 2.1 years younger, it’s a 2/3 chance the woman will outlive the man
- If the woman outlives the man, she has an expected 11.5 years as a widow (until she dies)
- If the man outlives the woman, he has an expected 8.8 years as a widower (until he dies)

For what it’s worth, not all these items are calculated on the same thing (they weren’t clear whether both man & woman were 65 years old, or if the man was 65 years old, and woman 2.1 years younger.

I’m going to calculate both (well, not exactly, but close enough for entertainment/educational purposes)

**EXPLAINING CONDITIONAL EXPECTATION**

If you want to learn about conditional probability, check out my post from 2000, which includes such insights as:

I could be going from info that was convincing me someone was female, to info convincing me someone was male! That’s why you’ve got to be careful of correlations: if I told you most people who had asses were female, and most people who had hips were female, but most people who had asses =and= hips were male, you’d think I was crazy. However, this is something that happens in real life all the time, due to all sorts of correlations. This is something called Simpson’s paradox.

Simpson’s paradox isn’t a paradox at all, by the way — it’s just something that illustrates people don’t understand conditional probabilities or weighted averages very well. Go to numbered page 119 in this (pdf page 129) for Martin Gardner’s explanation, which is the first such I ran into.

But in our case, it’s not correlations, but simply how expected value works when you throw in conditional probabilities.

I sketched out a case where there are two possibilities: event A occurring/not occurring.

Here is my hand-sketched equation:

And then here’s the math notation version:

What if there are multiple possible outcomes? Here’s the math notation version….

Okay, all that means is if you have a bunch of disjoint events A_0, A_1, etc. (disjoint means they don’t overlap — in probability terms, the probability of both happening is 0..aka mutually exclusive) — and the union of all these events is the whole probability space (i.e., their probability adds up to zero), to get the overall expected value, you take the weighted average of all the conditional expectations.

**USING THE NUMBERS FROM TIAA’S DOCUMENT**

Using, the prior link and numbers, if I want to figure out the average number of years a wife outlives a husband (whether or not she outlives him), then I can do the following:

Expected number of years wife outlives husband = Expected number of years if wife outlives husband * probability wife outlives husband – Expected number of years husband outlives wife * probability husband outlives wife

The negative is because number of years wife outlives husband is negative number of years husband outlives wife.

Using the numbers given, we have:

2/3*11.5 – 1/3*8.8 = **4.7 years**

Which is pretty damn close to the original 4.6 years one would guess.

**CALCULATIONS GALORE!!**

But I wasn’t satisfied. I wanted to calculate the probabilities for myself.

I found the Social Security mortality tables — there is no 2016 tables (yet) — I assume they used the 2013 Calendar Year tables from the 2016 Trustees Report… look, there’s not much difference between the two.

Because I’m doing this for fun, I decided to make my calculations really simple — I assume everybody dies at the beginning of the year (I can also do everybody dies at end of year for the other extreme, and then compare, but I will take this only so far.)

So I assume a male/female couple both age 65, and because of how I assumed people die at integral ages, there are three possible outcomes:

- Both die at same year

- Man dies before woman

- Woman dies before man

By the way, here are the probabilities, assuming independence of death probabilities (they aren’t in real life, but we’re not pricing joint last survivor life insurance, so let’s ignore that for right now):

- Both die at same year: 3%

- Man dies before woman: 57%

- Woman dies before man: 40%

Not the same as the prior info, but we’ll see when I redo for a 63-year-old woman and 65-year-old man

Here are the conditional expectations: 0 = Expected number of years woman outlives man, given they die the same year 11.17 = Expected number of years woman outlives man, given the woman outlives man (9.46) = Expected number of years woman outlives man, given the woman predeceases the man

And when we weight these appropriately, we get the woman outlives the man by **2.55 years** on average (which is averaging with when she dies before him)

**HEAT MAP OF JOINT DISTRIBUTION OF DEATH AGES**

Would you like to see what the probabilities look like? Here’s a heatmap:

(The max point is about woman age 88, man age 85)

**EXAMPLE WITH 63-YEAR-OLD WOMAN AND 65-YEAR-OLD MAN**

But let’s go back to the version TIAA has, and to simplify, I’m going to have a 63-year-old woman and a 65-year old man. Still using the 2015 calendar year Social Security tables, assuming death at beginning of year, and independence of deaths.

Here are probabilities:

- Both die at same year: 3%

- Man dies before woman: 62%

- Woman dies before man: 35%

Not exactly 2/3 and 1/3, but pretty close.

Here are the new conditional expectations:

0 = Expected number of years woman outlives man, given they die the same year 12.03 = Expected number of years woman outlives man, given the woman outlives man (9.25) = Expected number of years woman outlives man, given the woman predeceases the manThe overall expected number of years a woman outlives is then **4.18 years**. Close to the overall expectation one would think.

And here’s the new heatmap:

(Max is woman lives 25 more years… aka to age 88, and man lives 20 more years… aka age 85… just as before)

**REAL ADVICE ON RETIREMENT PLANNING FOR COUPLES**

That said, if you’re planning for retirement, one needs to think through all the discrete possibilities, because you are not living like a Schrodinger Cat – one spouse will outlive the other, and you need to think through what will happen.

When I bought term life insurance for Stu & me after our kids were born, I got more on him than me, actually, because of a variety of reasons. I recently bought additional life insurance on myself, permanent insurance, because now I have different probabilities of different outcomes that I have to plan for.

Some insurance products really don’t cost more when taken out on joint lives — for income annuities for couples, I highly recommend using a joint life annuity – some will reduce when first spouse dies, but it’s not like expenses necessarily drop for an elderly couple when one dies. The other may need to pay for services that the deceased spouse had provided (like driving around). And if most of your costs are fixed (housing costs, etc.), you really should get joint last survivor full options. Some will have options like paying 2/3 of the full benefit when one spouse dies.

Think of the peace of mind of having full payments continuing in exchange for the relatively small amount in payments upfront you’d be forgoing — I think it’s worth it.

In addition: when I was at TIAA, I made survivorship calculators for same sex couples as well (as I knew the TIAA customer base would likely have interest in such things) — even when the probabilities for each age are exactly the same, the surviving spouse can live several years past the deceased. It’s definitely true when there are disparate ages for spouses.

In any case, either surviving spouse may live several years after the other one dies. Plan accordingly.

Spreadsheet with calculations here. I may explain how to make the heatmap another time. :)

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Mortality Monday: The Sex Gap in Death

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