Burden

If you had $700 billion in $100 bills, how much would it weigh?

Approximately the same as all the people who ever voted for Sarah Palin.

Presidential terms and actuarial errors

Atlanta Journal-Constitution reports that the actuarial firm John M. Bragg & Associates has estimated the healthy life expectancy - average remaining life free of Alzheimer's or other impairments requiring an assisted living facility - for John McCain and Barack Obama. Bragg gives McCain a health expectancy of 8.4 years, and Obama 21.9 years, and offers the following conclusion:

“Either candidate can be expected to serve two full terms, without age or health being an issue,” said John M. Bragg, the firm chairman.

John Aravosis points out that Bragg's numbers don't add up:

The actuarial (sic) says McCain has 8.4 "healthy" years left. He mistakenly drew the conclusion that McCain would therefore be healthy through two terms in office, thinking that eight years from now is the end of McCain's term. That's incorrect. The end of the term is in 8 years 5.5 months from now. Statistically, that means McCain won't make it through two terms as a "healthy" president. By healthy, he means “the person does not require the care provided by an assisted living facility and is free of Alzheimer’s disease.”

By my back-of-the-envelope calculations, that means McCain wouldn't even be able to complete two full terms in office. His time runs out on December 30th or so of 2016, a good three weeks before the new president would be sworn in. So that means, statistically speaking, before the end of two terms, we should expect President McCain to have to be moved to a retirement home because he'll no longer be able to care for himself, and we should expect that he'll already be suffering from Alzheimer's.

Aravosis is correct in noting that 8.4 years from now does not give the new President (whoever it is) two full terms. His explanation of what that means is a little sloppy, but the end result is most likely correct. The issue is what it means that "we should expect" McCain (not) to be healthy and alive by the end of the two terms. I assume that Aravosis means "it is more likely than not", but that doesn't follow so simply from the published statistic.

Mathematical expectation is the theoretical mean, not median. What it means is the following: if we had a thousand men of the same age and other relevant characteristics as John McCain, we follow them all until they die and note how long each lives, and then we sum up all of their times from now until death, and divide the result by 1000 (the number of them), we will get something very close to 8 years, 146 days.

It does not mean that 500 of them will live longer than 8.4 years and the other 500 shorter. That would follow only if the remaining lifetime were distributed symmetrically, for example, if it followed the bell-shaped normal distribution. But that is not the case: all remaining lifetimes must be greater than zero, but there is no reason some of them couldn't be 30 years or more. The expectation (8.4 years) is obviously not in the middle of the range, so the distribution is skewed (asymmetric).

Now it turns out that, in most distributions of this kind (with a "tail" on the right), the median is less than the mean. If the median healthy survival is less than 8.4 years, it would mean that fewer than 500 of the initial 1000 "McCain clones" would have more than 8.4 years of healthy life - and, obviously, even fewer would have 8.46 or so years needed to complete two terms in office. So Aravosis made a correct statement, after all.

If Aravosis is correct, Bragg must be incorrect. It is not possible to say, based on Bragg's own numbers, that McCain "can be expected to serve two full terms, without age or health being an issue". Because the median is almost certainly lower than the mean, it would not be true even if the expected healthy life were somewhat longer, say 8.5 years, extending beyond two terms. It would still imply that McCain is more likely than not to die or become seriously impaired before the end of the second term.

But this error in arithmetic is not the most bothersome part of Bragg's statement. After all, if all the critique amounted to were nitpicking about chances being 49% rather than 51%, it would be rather academic. I see a far more serious problem in Bragg's choice of risk measure.

Suppose for a moment that Bragg's arithmetic was right, and that McCain could be expected, albeit barely so, to be alive and healthy on January 20, 2017. Suppose his chances are 55%. That's a generous "improvement" of Bragg's estimate, but it still means a 45% chance he would be dead or incapacitated. Would most people be comfortable with a president facing those odds? Would most people, upon hearing that number, characterize a potential McCain presidency as "without age or health being an issue"? I doubt that. From the risk management point of view, Bragg has cited an almost useless statistic.

John M. Bragg is not just any actuary. He was President of the Society of Actuaries. That, however, was a long time ago, when the US President was named Gerald Ford.

Dog bites man: miles per gallon edition

A Duke University study, published in Science and reported in the Consumer Reports, found that miles per gallon are misleading and that consumers would make better decisions if fuel consumption were posted in gallons per mile.

Well, duh. Shouldn't that be obvious?

Is it any surprise that people calculate better with direct quantities than with inverses? Miles per gallon is an inverse measure of the cost of fuel, so one doesn't have to be a rocket scientist to figure out that consumers will have more trouble making rational decisions based on that measure than based on a direct measure of fuel consumption, such as liters per 100 kilometers, which is used in Europe and pretty much the rest of the world.

Most people don't realize (at least not immediately) that driving 45 mph from Startington to Endington (distance: 60 miles) and 80 mph on the way back takes more time than driving 60 mph both ways. But they do realize that a 80-minute trip and a 45-minute trip together take longer than a two-hour trip. Similarly, people don't realize that going from 12 to 16 mpg is a greater saving than going from 24 to 32 mpg, but they would recognize immediately that going from 20 to 15 l/100 km is more significant than going from 10 to 7.5 l/100 km. Elementary, my dear Watson!

Thou Shalt Not Extrapolate Exponential Growth Rates

If you think housing in the DC area is expensive now, just wait 50 years:

The findings, discussed at a forum sponsored by the Metropolitan Washington Council of Governments, show that the price of the average home in the area will grow to $14 million in 2057 from today’s average of $477,000.

Should we take this seriously? After all, I saw it in a free newspaper. But it quotes a real study, and its author, a professor at George Mason University. Let's read a little more:

Researchers predicted, however, the average household income would only climb to $1.3 million from today’s $137,000 during the same period.

“We already have an affordability problem,” said George Mason University professor Stephen Fuller, who calculated the estimates. “But this is really scary. It is going to take 11 times the average household income to afford the average-priced house.”

Wait a minute. Who will buy those houses? At 6% annual interest rate, a 30-year mortgage on an average home would require payments exceeding three-quarters of an average household's income. And that's just mortgage payments - no property tax, insurance, maintenance, or utilities. Even at the entirely unrealistic zero interest, mortgage payments would equal 36% of income. Prices can be astronomical in a limited area, where only the region's elite can afford to live, but the study refers to the entire Washington Metropolitan Area, which, according to the article, will then span from Baltimore to Richmond and have a population of 9.9 million. If only the top 5-10% can afford housing in the area, where will the other 9 million live? There is a violation of the law of supply and demand in the study's results.

That study is yet another example of the dangers of extrapolating exponential growth rates over long periods. If you calculate the annual growth rates implicit in the study's results, they look reasonable: housing prices increasing 7% per year, and incomes 4.6% per year. Assuming 3% inflation, the real (net of inflation) rates are 4% and 1.6%. I might use those numbers myself if I had to predict prices and earnings 2-3 years from now.

The problem is that the difference in growth rates of 2.4% compounds to a factor of more than 3 (in other words, a difference of over 200%) over 50 years. What makes sense over short periods doesn't necessarily make sense in the long run.

For a related example, consider health care costs. Their share of GDP grew from 7.2% in 1965 to 16% in 2005, at an average annual rate of 2.7%. If that trend continued for another 50 years, health care would account for 60% of GDP in 2055. Keep in mind that health care is itself a sector of the economy, so its product is counted in the GDP, and if it makes up 60% of GDP, it means that it is one-and-a-half times as big as the rest of the economy combined. If that doesn't make the absurdity of the result obvious, try to ponder what would happen just 8 years later, when health care expenditures would exceed 100% of GDP.

Similarly, if you looked at the rate of growth of market capitalization of Nortel or Enron in the years before their respective stocks crashed, and extrapolated those rates into the future, you would conclude that those companies would own everything in North America at a time not too far into the future.

Or take population groups. Population growth in Muslim countries has been higher than the world's average, so Islam has been increasing its share, which currently stands at about 19% of the world's population. According to some estimates, the number of Muslims has been growing approximately 0.6 percentage points faster, annually, than the world's population. At that rate, by the end of 24th century, everyone in the world will be Muslim. The only problem is, Pentecostalism, with about 2% of the world's population as its adherents, is growing about 6% faster than the world's population, so by 2075, everybody in the world will be Pentecostals.

A more cheerful extrapolation shows that in just 10 years, every computer in the US will be a Mac.