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The Argument About Time Diversification

Should time horizon affect asset allocation?

John Rekenthaler, 12/01/2015

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 if 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.

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.)

is vice president of research for Morningstar.

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