The Argument About Time Diversification

Say What? It seems so straightforward: Because stocks are more volatile than bonds, and bonds more volatile than cash, an investor's asset mix will vary according to time horizon. Preparing for next year's house down payment? Mostly cash, perhaps a few short bonds, probably no stocks. Saving for a 2055 retirement? Mostly stocks, some longer bonds, little cash.

The economist Paul Samuelson begged to disagree. In 1963, he published a very short but widely cited paper entitled Risk and Uncertainty: A Fallacy of Large Numbers. In the article, Samuelson attacked the notion that a single undesirable gamble can become a desirable gamble if it is made in numbers.

Consider the gamble that a coin lands heads, you are paid $1,200. If it lands tails, you must pay $1,000. Many people would respond that they don't fancy that particular gamble as a single offering, but they would be happy to accept the offer if the gamble were offered 100 times. They will win the single gamble only 54.5% of the time, but, because of the law of large numbers, they will win the 100-gamble wager 97% of the time.

Samuelson quarrels with both the intuition and the conclusion. Yes, he says, the odds of losing decline as the number of bets rise. But when the bettor does land in the red, the losses can be greater. After all, the single gamble can lose no more than $1,000, while the 100-gamble sequence can conceivably cost up to $100,000. Magnitude counts. (I could have written that differently, but it's a tired joke.) Therefore, repetition of the bets does not wash away the risk.

Extending the argument from coin flips to asset allocation was a natural step. Stocks, too, are a risky gamble, and the standard belief in asset allocation is a form of the law of large numbers, that stocks become relatively safer over time because of repetition. Sometimes stocks win, sometimes they lose, but as time passes and there are more draws from the stock bucket, the probability increases that stocks will not only be absolutely profitable but also relatively more profitable than bonds. In 1969 and again in 1971, Samuelson disputed that logic. Stocks may fall short less often as the time horizon grows, but the amount of the shortfall can be much larger. Thus, stocks do not become safer over time, and time horizon should not be a factor with asset allocation.

While Samuelson's articles were respected by academics and institutional investors, they did not change much investment behavior. Today, as a half century ago, financial advisors ask their clients about time horizon. The Chartered Financial Analyst program includes Samuelson in its curriculum--but it also instructs prospective CFAs to consider time horizon when making an investment proposal. Even fellow professor Jeremy Siegel, at Wharton, counsels "stocks for the long run." Samuelson spoke--but apparently, nobody listened.

Counterpoints One reason they did not is because of the simplifications of Samuelson's model. His conclusions derive from the expected-utility function, first developed by Daniel Bernoulli in the 18th century. Bernoulli's model, as with most useful economic constructs, is streamlined. It does not address human capital--the ability of a young person to compensate for investment losses by working harder and earning more money. Nor does the model consider real, tangible constraints that may prevent the investor from absorbing more than minimal short-term losses, such as the requirement to pay a known liability.

More broadly, in an observation that helped to earn Daniel Kahneman his Nobel Prize, expected utility does not use reference points. The model looks at gains or losses in isolation. In contrast, people judge their results while looking at where they had been. Context counts.

(And of course, Bernoulli's math cannot account for the human element. Investment managers can and do get fired for trailing their peers because they followed the advice of quirky academics.)

These critiques, while damaging, do not topple Samuelson's central tenet. Yes, these caveats may prevent investors from acting fully rationally according to the expected-utility model, but the logic remains. Absent other factors, as the number of bets grows, the increasing size of a possible shortfall offsets the rising probability of a gain. Per Samuelson, the math is inevitable.

The assumption that underlies his math, however, is not. There, it seems, Samuelson's argument fails. The expected-utility framework appears to collapse when the number of bets becomes so large that the probability of success approaches 100%. In those cases, the model draws conclusions that do not look to be "fully rational." With high-probability events, the expected-utility function neither describes how people do behave, nor, critically, how they should behave.

Let's return to our previous gamble. Heads you win $1,200, tails you lose $1,000. Perhaps you would accept that bet; perhaps you would not. If the offer is extended so that it becomes that gamble repeated 800 times, with no money being exchanged until all the gambles are completed, you would have a 99.99999% chance of landing in the black. Samuelson states that for a given level of risk tolerance, that wager is no more or less attractive than the original wager. I suspect that you disagree.

To put the matter another way: If somebody offered you the chance to receive an average expected payout of $80,000, with one chance in 10 million that you would owe money instead of receiving a payout (with most of the negative cases being losses of less than $80,000), would you take the offer? You betcha.

In fact, you probably accept a significantly less attractive gamble each day. The U.S. automobile accident rate is roughly 1 death for each 100 million miles traveled. Thus, a drive of 10 miles equates to one chance in 10 million of dying in a traffic accident. A 10-mile drive, therefore, offers the same odds as does the gamble of the 800 coin tosses--one bad outcome (we'll set nonfatal accidents, speeding tickets, and running out of gas to the side) in 10 million tries, with the rest being good outcomes. Except that the payoff for the drive is whatever 10 miles in a car is worth rather than the $80,000 of the coin flips, and the shortfall consists of your life, not your money. The drive is a much worse proposition than are the coin flips.

In short, if you accept Paul Samuelson's logic on asset allocation, then you had better stop driving except for extreme emergencies (or if very large payouts await at your destination).

(Harvard's Matthew Rabin states the objection more formally--"Expected-utility theory seems a useful and adequate model of risk aversion for many purposes, and it is especially attractive in lieu of an equally tractable alternative model … But this and previous papers make clear that expected-utility theory is manifestly not close to the right explanation of risk attitudes over modest stakes.")

On Deck If you remain unconvinced by the dissents and believe that asset allocation should indeed be independent of time horizon, then you will wish to skip the next column, which addresses the following question: If it is acceptable to vary asset allocation according to time horizon, then how long must that horizon extend for stocks to be regarded as relatively safe choices?

John Rekenthaler has been researching the fund industry since 1988. He is now a columnist for Morningstar.com and a member of Morningstar's investment research department. John is quick to point out that while Morningstar typically agrees with the views of the Rekenthaler Report, his views are his own.