When does the case for long-term investment make sense?

Paul Samuelson showed why time horizons matter less than commonly thought

ONE LUNCHTIME around 1960 a professor proposed a wager to a colleague. Flip a coin and call “heads” or “tails”. If you call right, you win $200. If you call wrong, you pay $100. This is a favourable bet for anyone who would take it. Even so, his colleague refused. He would feel the loss of $100 more than the gain of $200. But he would be happy, he said, to take 100 such bets.

The professor who offered the bet, Paul Samuelson, understood why it might be refused. A person’s capacity for risk could no more be changed than his nose, he once said. But he was irked by his colleague’s willingness to take 100 such wagers. Yes, the likelihood of losing money after that many tosses of the coin is vanishingly small. But someone who takes very many bets is also exposed to a small chance of far bigger loss. A lot of bets, reasoned Samuelson, were no safer than a single bet.

This lunchtime wager was of more than academic interest. It drew the battle lines in a debate on the merits of long-termism. Samuelson challenged the conventional wisdom that his colleague embodied. In later work, he used the bet as a parable. He showed that, under certain conditions, investors should keep the same fraction of their portfolios in risky stocks whether they are investing for one month or a hundred months. But what Samuelson’s logic assumed does not always hold. There are cases where a long-term horizon works in investors’ favour.

To understand the debate, start with the law of large numbers. It means that the more often a favourable gamble is repeated, the more likely it is that the person who takes it comes out ahead. Though a casino may lose on a single spin of the roulette wheel, over a large number of spins its profits are determined by the slight advantage in odds (the “house edge”) it enjoys. But a casino that would take a hundred $100 bets would not refuse a single bet of the same size. That was part of Samuelson’s beef. If his colleague dislikes a single bet, after 99 bets he should refuse the 100th. By this logic he should also refuse the 99th bet, after 98 bets. And so on until all bets are spurned.

Clouds on the horizon

Only a naive reading of the law of large numbers would support a belief that risk is diminished by more bets, said Samuelson. The scale of potential losses rises with the number of bets. “If it hurts much to lose $100,” he wrote, “it must certainly hurt to lose 100 x $100.” Similarly, it is foolish to believe that by holding stocks for the long haul—taking multiple bets on them—you are sure to come out ahead. It is true that stocks have usually yielded higher returns than bonds or cash over a long period. But there is no guarantee they will always do so. Indeed if stock prices follow a “random walk” (ie, an erratic and unpredictable path), long-term investing holds no advantage, said Samuelson.

This logic begins to fray if you relax the random-walk assumption. Stock prices appear to fluctuate around a discernible trend; they have a tendency, albeit weak, to revert to that trend over very long horizons. That means stocks are somewhat predictable. If they go up a long way, given enough time they are likely to fall, and vice versa. In that case, more nervous sorts of investors are able to bear a higher exposure to stocks in the long run than they would be able to in the short run.

Samuelson’s reasoning also assumes that people’s taste for risk does not vary with how rich or poor they are. In reality, attitudes change when a target level of wealth is within reach (say, to pay for retirement or a child’s education) or when outright poverty looms. When such extremes are far off, it is rational to take on more risk than when they are close. The calculus also changes with a broader reckoning of wealth. Young people, with decades of work ahead, hold most of their wealth in “human capital”, their skills and abilities. This sort of wealth is a hedge against riskier kinds of financial wealth. Indeed the more stable a person’s career earnings are, the greater the hedge. It follows that young people should hold more of their wealth in risky stocks than people who are close to retirement.

Samuelson vigorously disputed the dogma of long-termism, which says that the riskiness of stocks diminishes as time passes. It doesn’t. That is why long-dated options to insure against falling stocks are dearer than short-dated ones. The odds of winning favour risk-takers over time. But they are exposed to big losses in the times when they lose. Still, it would also be dogmatic to say that time horizon does not matter. It does—in some circumstances. What Samuelson showed is that it matters less than commonly thought.