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Invest $100,000 in a typical retirement account when you are 25, and how much should you expect to have when you collect your gold watch 40 years later

Based on some standard industry assumptions about investment portfolios, returns and volatility, a money manager might tell you that the "average" outcome would be in the region of $1.2 million, adjusted for inflation.

But here is the problem: That "average" masks a lot of trouble. Even if you use those same standard assumptions, more than half the time you will end up with less than $750,000. And a lot of the time you will end up with less than $350,000. Contrary to popular myth, the results aren't distributed normally, in a bell curve. Thanks to volatility, they are bunched at the low end.

From $1.2 million to $350,000 or less -- that is going to make quite a difference.

This mathematical analysis appears in "Investing for Retirement: The Defined Contribution Challenge," a new paper by Ben Inker and Martin Tarlie at GMO, a Boston-based investment firm with $117 billion under management.

It took me back to my high-school math, and the difference between the three types of average -- the mean, the median and the mode. To put it in a nutshell, that $1.2 million is the mean, or simple average, and it pretty much is useless for planning purposes.

Few financial issues are more important to us all than planning for our retirement. And yet our approach to the subject is riddled with some serious logical and mathematical errors, Inker and Tarlie argue.

We misunderstand averages, volatility and the likely distribution of returns, they say. And we may be misunderstanding financial history as well.

Many retirement planners, including the people from the 401k company hosting the free seminars at your workplace, derive their forecasts for future returns from the past. That is perfectly understandable, as far as it goes. But it is flawed.

Data from New York University's Stern School of Business show that since 1928, U.S. stocks, as measured by the S&P 500 ($INX) have earned average returns of 9.6 percent a year, while 10-year Treasury notes have earned 5 percent a year.

But today the 10-year Treasury sports a yield of just 2.8 percent. Good luck trying to squeeze 5 percent a year from a 2.8 percent bond, Inker notes. Meanwhile, the S&P 500 now trades at around 25 times cyclically adjusted per-share earnings of the past 10 years, a measure known as the Shiller price/earnings ratio, after Yale finance professor and Nobel laureate Robert Shiller.

Yet data tracked by Shiller going back to the 19th century has found that when the U.S. index has traded at these levels, subsequent returns have typically been dismal -- often failing even to keep up with inflation.

"This is a dangerous time to be close to retirement," said Inker, co-head of GMO's asset-allocation committee, when we met earlier this month in the firm's offices overlooking Boston Harbor. Because of elevated asset valuations, relying on standard planning may "leave you in serious trouble."

So what can we do about it?

Some of the authors' conclusions aren't controversial. Investors need to get a better understanding of the risks of poor returns. And they need to save more -- often much more -- to compensate.

Two of Inker and Tarlie's other conclusions may raise eyebrows. First, and counterintuitively, they say that many investors may need to take the plunge and invest more aggressively than conventional wisdom suggests. That means holding more stocks and fewer bonds. That is because, even though stocks are volatile and returns may be well below historic averages, over time they still tend to beat bonds by a wide margin. And many investors simply can't afford low returns.

Second, they argue that investors should hold more foreign stocks, as opposed to U.S. stocks.

GMO believes international stocks are a better value than those in the U.S. and offer better returns. But the argument stacks up even if you don't necessarily embrace that analysis. If we should diversify as much as possible to earn the best returns for the risk, shouldn't we invest globally as a matter of course? Why should we have most of our eggs in the S&P 500?

At a social gathering, I once met the head of "target date" retirement funds at a big investment company. I asked him why, across the industry, all such funds -- in which allocations grow more conservative over time -- were so overinvested in U.S. stocks at the expense of global diversification.

"Because that's what the clients want -- it's what we can sell," he confessed. It had nothing to do with what was actually likely to be best for the clients in the long run.

A spokesman at the Investment Company Institute, which represents the mutual-fund industry, noted that international stocks, including emerging markets, accounted for 30 percent of the equity allocation of the typical target-date fund. But this still is well below their share of the global stock-market indexes or global economic output.

It is hard to see why the default allocation shouldn't be to put one third of one's equity allocation in U.S. stocks, one third in developed overseas markets and one third in emerging markets -- even if you aren't taking a view about which is better value.

GMO has a reputation for taking a deeply cautious view on markets. But that doesn't undermine the analysis by Inker and Tarlie. And it leads to one other implication that they don't pursue but which I can't avoid raising. If investment returns from stocks and bonds look so meager today, it may be sensible to park more money on the sidelines in cash or an equivalent, such as a money-market or stable-value fund -- and wait, and hope, that stocks or bonds will get cheaper.

Is that "timing" the market? Yes. Is it "risky"? Yes.

But so is trusting the "averages."

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