Loss dispersion

This column explains why investment practice diverges from his theory. Let’s begin with Samuelson’s logic. His assertion is founded on the mathematical truth that while the annualized rate of return for risky assets converges over time—the stock market being likelier to lose 20% in a single year than 20% annualized for a decade—the opposite occurs with investment dollars. For example, a 3% annual loss over 10 straight years leads to a 26% overall portfolio decline.

From there, Samuelson proceeds to another mathematical truth. According to the expected utility hypothesis, which describes how people should make rational decisions, this dispersion in dollar returns negates the advantage that comes from holding stocks for the long run. That is, although the odds that a stock portfolio will be profitable improve over time, so does the size of the shortfall should they fail. The danger occurs less often but more severely.

The caveats

Several assumptions that underlie Samuelson’s argument have been criticized for being unrealistic. One complaint is that his formulation does not assume that stocks revert to the mean, so that prolonged stretches of poor returns are to an extent rescued by rallies. Another is that it does not consider human capital. Workers who lose money on their long-term investments are better off than the computation avers because they can earn back some of those assets.

I will set those challenges aside. They are consequential but for the purposes of this column irrelevant. In my experience, investors do not shun Samuelson’s recommendation because they believe that stocks are mean-reverting. (Quite the contrary; most become cheerier after stocks have risen and more pessimistic after they have declined.) Nor do many young workers think that if the stocks in their 401(k) accounts flop, the failure is acceptable, because they can fix the deficit by working until age 75.

When risk looks equal

There are, however, two substantial objections. One is that the expected utility function does not always correlate with how people view risk.

Sometimes, it does. For example, in his discussion of Samuelson’s work, Mark Kritzman shows the potential outcomes for $100 placed in a simplified portfolio that either gains 33% or loses 25% each year, with equal probabilities.

Test case

In this instance, my intuition agrees with the expected utility function. I am no more enthralled with the prospect of gaining either $33 or losing $25 over one year than I am of either the two- or three-year investments. No question, realizing $237 with the top-scoring result for the three-year period would be pleasant, but offsetting that opportunity is an equally sized probability of a 58% loss. Ouch!

When it does not

Kritzman’s example, however, is a poor investment. Its expected geometric return is zero, while its annual standard deviation is 29%. Unless such a security provided outstanding diversification, you would never knowingly purchase it. Let’s see how the numbers look with a more appetizing case. Over the past century, U.S. equities have returned an annualized 7%, after inflation, with 19% standard deviation. What are the 99th percentile, median, and 1st percentile totals for such an investment over one year and 30 years?

Here are those amounts, courtesy of Morningstar’s James Xiong, who graciously provided the calculations. (Thanks, James!)

Hmmm. Not only are the 30-year payoffs higher across the board for the 30-year period, but because those returns are calculated in real terms, that 1st-percentile result of $88 isn’t all that bad. If the risk-free alternative matched the inflation rate—which is roughly what has occurred over history—the 30-year risk-free return would be $100, only a modest improvement on the 1st percentile equity total.

With these inputs, the 30-year payoff does not trail the one-year result until reaching the 0.5% percentile, meaning one instance in 200. For most investors, that risk will not dissuade them from preferring the 30-year wager. But the version of the expected utility hypothesis used by Samuelson regards the two horizons as equally attractive. In fact, it does so no matter what forecasts are used for equity returns and volatility.

Verdict: Investors almost universally are irrational, according to conventional economics. Oh well. If preferring that 30-year wager to the one-year choice is irrational, I am delighted to stay that way. I strongly suspect that you are, too.

What about time buckets?

The other substantive quarrel with Samuelson’s argument is provided in this note from the Society of Actuaries. The author argues that the expected utility derived from a series of wagers, as analyzed by Samuelson, is merely a single, narrowly defined way of evaluating financial risk. Another perspective lies outside Samuelson’s analysis, that being workers’ need to accumulate sufficient assets for retirement. Doing so requires that their investments exceed the risk-free rate.

In contrast, workers have no great need to earn above the risk-free rate when funding shorter-term goals, such as making a down payment on a house or buying a car. The difference in portfolio values between making an annualized 8% for three years, as might come from stocks, and the 3% they could achieve with cash is minimal. What overwhelmingly matters for footing such bills is not investment returns, but instead the amount that is saved.

In other words, even if the expected utility formula stipulates that the conditions are similar regardless of time horizon, the investor’s needs are not. There is greater reason to court investment risk for the long term than for the short term.

This objection strikes me as fully warranted. Even if one grants the infallibility of the expected utility function for judging the merits of investment bets, that formula cannot determine why one might invest. Such considerations lie outside the scope of Samuelson’s analysis, as I suspect he himself would have granted.

In summary

Equities do improve with age. Samuelson’s argument is intriguing; I quite enjoyed working through it. And the insight that the potential dollar amount of investment losses increases over time is worth remembering. Stocks for the long run are not a fail-safe proposition! Ultimately, though, I must side with the collective wisdom: Risky assets are best suited for the patient.