Why Invest? - Part 2

For although risk tolerance is mutable, it surely shouldn’t whip this way and that with every flutter of the market or news cycle. I often hear investors say, in response to changes in market prices and frightful or cheerful news reports, that they are becoming “more conservative” or more “aggressive.” These locutions may be all right if they actually mean that, for one reason or another—a good reason, I hope, based on an economic outlook—the investor has changed his portfolio to give it a more conservative or a more aggressive slant with respect to prospective return and risk. But the locutions subvert the definition of risk tolerance if the investor means that he can handle risk only when he doesn’t expect anything bad to happen. If that’s what he means, then, rather as the Oscar Wilde character declares that he can resist everything except temptation, the investor is implying that he can face every risk except the prospect of a loss.

But risk tolerance may very well change slowly over time. For an organization with an endowment, it may change as the personnel on the board’s executive committee change, or as the mission of the organization changes. There is also a common belief that for individuals, risk tolerance either will or should decline as age advances. Investment professionals advise individuals to adjust their portfolios accordingly. The reason for this belief, sometimes unstated and usually unexamined, is that investment risk declines as the time horizon lengthens, because, on average, large bad returns will tend to be cancelled out by large good returns as time passes. And because investment risk declines as time lengthens, you can be more tolerant of it if you have a long time horizon. This explanation is so commonplace that it may seem perverse to question it. But question it, we will.

The time diversification puzzle

Paul Samuelson, one of the preeminent economists of the twentieth century, published a landmark paper in 1963² that would have raised the hackles of investment advisors if any had read it and had hackles, because he demonstrated that this conventional idea is a fallacy. The contradiction between the common notion (that risk tolerance increases with one’s time horizon) and what Samuelson demonstrated became known as the time diversification puzzle, where “time diversification” refers to holding an investment over many periods of time, similar to the way that “diversification” normally refers to holding multiple investments during any given period of time. (Samuelson described his work in financial theory as his “Sunday painting,” a pleasant part-time recreation, but although his reputation rests on his work in the wider field of economics, his financial studies, which he kept up to the end of his life, significantly advanced the subject and are not entirely separable from his other contributions.) Samuelson proved that, although the conventional statistical measure of investment risk does, indeed, decrease with time, an investor is financially no better off because of this, if three reasonable assumptions hold. These are, first, that changes in wealth over time are entirely the result of investing, second, that investors don’t change or allow a change in their investments’ balance between return and risk, and third, that the investments’ return in any period does not depend upon their return in an earlier period.³ (This last means that there is no reversion to the mean in the behavior of the investments—a point that is debated by those who have made a careful analysis of historical numbers.)

Say we expect the annual rate of return of the stock market to be 7% a year,⁴ and our expectation of volatility is 20% per year. (Never mind what that actually means; as usual, I’m applying the common statistical definition of volatility and its rough value, which we know from stock market history, and as I’ve said before, we also know that it underestimates the probability of extreme bad results.⁵) Using these numbers alone and a standard statistical table or a spreadsheet, I can find that, over any one year, there is a probability of just under 1-in-3 that an investment in the stock market will suffer some loss, of whatever size. Here is a table of the probabilities of losing at least some of your initial investment in the stock market, for an investment held over increasing time horizons:

At this point, you don’t see a problem. The probability of losing money in the stock market is decreasing as the time horizon lengthens. That is, your risk seems to be going down.

But now consider this table. It shows how much of your initial investment you could lose, with a probability of 1-in-100, at some point during each of those same spans of time:

This means, for example, that if you hold your investment for ten years, there is a 1-in-100 chance that you could lose at least 65% of your initial amount at some point during that time.

Seen this way, your risk is actually increasing as time lengthens, which was Samuelson’s point.

If this perplexes you, and you feel that it can’t be right, it’s not just because you’re not an economist. The puzzle has slipped up even the mathematically adept. The mathematics of probability are often discordant with common sense.⁶

Simply saying, “But I can wait to the end,” doesn’t resolve issue, first, because there is very seldom a definite end and (as can be shown mathematically) a rational investor will find that these increasing potential losses render the investment, if no worse, then certainly no better than the investment held over short time horizons, and second, because few investors hold their investments intact over their time horizons. Say you’re investing for retirement. You’re going to make withdrawals over time, and if you have to make a withdrawal after a major loss in the stock market, you’re going to have much less to invest for the future. There are many retirees or would-be retirees today who learned this lesson in 2000 or 2008 despite having long time horizons.

Coming to terms with Samuelson’s puzzle requires our acknowledgment that we have two definitions of risk, one in terms of volatility, the other, which I think more suitable for individual investors, in terms of the probability of loss of wealth, or rather, the loss of wealth that is possible for a given probability. As a matter of mathematics, these aren’t inconsistent definitions, and one can translate between the two. Those with a fundamentalist commitment to the statistical identification of risk with volatility will argue that risk does decline over time, in that there is a tendency for the ups and downs of the market to cancel each other. But if you convert investment risk into the magnitude of a loss for a given probability, which is the practical concern of most investors, then it does not decline over time; the dispersion of possible outcomes increases.

That seemingly leaves in shards the common notion that we can take on more risk the longer our time horizon. Can we put it back together? Yes, because the resolution of the puzzle lies in our seeing this investment conundrum in a larger economic context, in which case two of Samuelson’s three assumptions no longer hold. But the explanation of this solution must await a later essay, when I will consider portfolio construction for the individual investor. Let no one persuade you, though, that you should be more tolerant of risk only because you have a long time to invest. Your risk tolerance very likely will, indeed, decrease over time, but you can scarcely expect to derive its correct value from an incorrect principle.

Risk tolerance and the non-rational investor

The inquisitive reader who browses the occasional investment-related article in the popular press or a science magazine may have wondered why, except for a brief dismissal in my first essay, I haven’t said more about that beguiling subject, behavioral finance. I have consistently been looking at investments as a rational actor would or should. But the concept of risk tolerance is one of the main points of contact between investing and the limited rationality that is the subject of behavioral economics.⁷ Risk tolerance is largely psychological no matter how you look at it, whether it is rational or not. One of the earliest insights of behavioral economics was codified by Daniel Kahneman and Amos Tversky in what is known as Prospect Theory, which describes risk tolerance in a way very different from what has underlain my account to this point. It can be expressed mathematically, but its gist is that, with respect to an absolute reference level of wealth, an individual values small gains proportionally more than large gains, and fears small losses more than large losses, though losses in general hurt more than gains feel good. Described this way, the theory may seem innocuous and a fair account of human behavior. But it is easy to demonstrate that it describes irrational behavior. Here’s an example that shows why:⁸

Imagine that you face the following pair of concurrent decisions. First examine both sets of choices, then decide which option you prefer for each.

Choice B is worth $2,500 (= (1/4 * $10,000) + (3/4 * 0)), so a little more than Choice A, but most people prefer A to B, because it’s without risk. Choice D is worth -$7,500 (= (3/4 * -$10,000) + (1/4 * 0)), so the same as Choice C, yet most of the same people prefer D to C. That is, at one and the same time, they would in one case give up some value in order to avoid risk, and in the other case, where the amounts of money at stake are larger, they’d take on risk, though the value is the same as the alternative choice. By the tenets of economics, these are not coherent, rational decisions.

While I continually make the point that, in the end, it is wealth, not return (which is relative change in wealth) that matters, you may have noticed that until we looked at the time diversification puzzle, wealth was mostly absent from our discussion of risk tolerance. That’s because, by considering volatility as a measure of risk, we can translate risk directly into probabilities of loss. Yet Prospect Theory codifies the observation that not only do investors behave as if wealth does indeed matter in consideration of risk, but more curiously (from the standpoint of Utility Theory, which I introduced in Part I of this essay), the amount of wealth of the status quo ante, and the measure by which some might be gained or lost, affects the subsequent decision.

But Prospect Theory is a description or generalization of how people actually make decisions when faced with risk, rather than a normative economic model derived from first principles. It shouldn’t change the theory of how one ought to invest. Rather, it sums up a psychological process that an investor may have to overcome on the way to identifying his or her risk tolerance for the purpose of making rational economic choices. We ought to invest to get the best value after consideration of risk. Perhaps Prospect Theory offers the kernel of an explanation of the unstable expressions of risk tolerance I described earlier. If the theory provides an accurate account of how the investor feels about the risk and return of investing, then it implies, too, that even when we observe how an investor behaves in reaction to actual changes in the value of his investments (as distinct from the hypothetical changes that a risk tolerance questionnaire may offer), we are still incapable of deducing correctly what the investor’s risk tolerance actually is or ought to be. The investor may not be behaving rationally. This is all the more justification for an investment advisor to intervene and to coax out of the client a more rational expression of risk tolerance, or for the self-directed investor to engage in a bout of introspection, informed by statistical data and a knowledge of how the mind can subvert economic self-interest in the face of risk.

The utility of wealth

A while back, we came to a fork in the trail toward the estimation of risk tolerance and took one path, which led us through the thicket of probability theory until we arrived in the general vicinity of our destination. Let’s now return to that junction and take the alternative path, which rises above probability to an overview of wealth and utility before descending again to an estimate.

By way of reconciling the identification of risk with, on the one hand, volatility of returns, and on the other hand, with a shortfall in the amount of money desired, I’ve said that volatility can be translated into the probability of a shortfall (and always added the necessary proviso that volatility does this very imperfectly, giving a false sense of only mild insecurity regarding extreme shortfalls). This implies that it doesn’t much matter which of the two definitions of risk you choose. Yet despite this, we now find that the two definitions present us with differing interpretations of tolerance. This apparent inconsistency results from a myopic focus on risk and risk aversion, and not the whole picture of the investor’s well-being, a concept that theoreticians have long identified as utility. Utility takes into consideration both the level of wealth and changes in that level. It was this concept that Daniel Bernoulli introduced.⁹ There is monetary value—and then there is utility, which is what that monetary value is worth to us. For example, the prospect of a $10 gain or loss matters more to a beggar than to a hedge fund manager. And because risk can be understood as the probability of changes in the level of wealth, it is a constituent of utility.

But utility is no more measurable than is risk tolerance or risk aversion: You can’t directly measure how much more that $10 is worth to the beggar than to the hedgie. Utility Theory does, however, explain why there is a difference and what it ought to be, and it was in opposition to Utility Theory that Prospect Theory was formulated. But whereas Prospect Theory is only descriptive, Utility Theory is both descriptive and prescriptive. Even in falling short of the mark as a description of human behavior, it still can’t be beaten as a prescription for it. That’s why sound investment decisions rest on the foundation of Utility Theory.

To give numerical flesh to our spectral visions of the utility of our investments, we can harness the power of computers to project future values. With Monte Carlo simulation, computers can fabricate the behavior over time of investments and their interactions with our planned expenditures and savings. In an earlier essay, I have already briefly described how Monte Carlo simulation works.¹⁰ The point of it, though, is that starting from our expectations of return and the probabilities of different returns, and our planned additional investments and expenses, computers can calculate not just the range of possible values of our wealth at a chosen time in the future, but also, for each value, the probability that we will attain at least that much. Figure 1 is an idealized and simplified version of what a Monte Carlo simulation might project for a date in the future, say, 20 years from now. We’re assuming that you’re starting with $800,000, you’re saving $15,000 every year, you’ve also specified your planned expenditures and their dates, and you’ve balanced the likely return and risk of your investments in such a way that we expect their annual rate of return to be 5%.

The chart represents a slice at one point in time, twenty years hence, through the range of possible outcomes. The height of the curve corresponds to the probability of attaining a particular value shown on the horizontal axis. The vertical line marks the median outcome: there’s a 1-in-2 chance that you’ll end up with more, and a 1-in-2 chance that you’ll end up with less (and it’s a matter of statistics that the median value is greater than the value of the most probable outcome). We can create such slices for different times in the future. By varying the investment choices, re-running the simulator, and judging the range of possibilities and their probabilities of wealth at different points in time, an investor can (at least in principle) choose the set of investments that results in outcomes that are of greatest utility to him, and we can work backwards from these investments to the present risk tolerance that they imply.

In the last decade, simulation has become a popular tool among investment advisors. The projections of future wealth, however, suffer from the limitations of the simulation methods, which, unbeknownst to the users, may rest on unrealistic models of investment risk; in particular, they may underestimate the probabilities of extreme bad results. There may be other problems, too, like very inaccurate estimates of the effects of taxes and the costs of buying and selling investments, and the system may even fail to allow for changes in your risk tolerance over time. In nearly all cases, the innards of the software are proprietary, and even if the vendors were completely forthcoming, only an expert, and not the typical user, would be in a position to diagnose them. Furthermore, even if the simulation is good, the intended beneficiary of the process, the investor, may not easily grasp the representations of the results. Those of us who are “quants” too easily assume that those who aren’t can grasp intuitively the probabilities of loss. Again, there must be a conversation about the results, even if you lack an advisor and you’re talking over your investments only with yourself.

Conclusion

Should you choose to invest, and to the extent that you invest, you inevitably expose yourself to more than a minimal risk of loss. You invest with both the hope and the expectation of an increase in wealth, but you must face the possibility that reality may not live up to the expectation; indeed, you may end up worse off than if you had not invested at all. (Then again, it’s also possible that you will exceed your expectations.) The full range of possible prospects, and their varying probabilities, matter intensely to your wellbeing. Upon them depend the likelihood of your funding your retirement, maybe your health insurance, your children’s college education, and the pecuniary legacy you want to leave. You must consider not only whether your investments are good ones, but also how best to balance a reasonable expectation of their returns against the risk of harm that ensues from chasing gain.

Investments, either single stocks and bonds, or bundles of them, like mutual funds and exchange-traded funds, should be thoughtfully aggregated into portfolios that suit each investor’s tolerance for investment risk, or, what amounts to the same thing, aversion to it. Every pre-packaged bundle of investments—a mutual fund, a hedge fund, a private equity fund, an ETF—reflects, either by design or by default, the return/risk tradeoff chosen by its manager. You can regard this as its manager’s risk tolerance, but the manager is the agent for his clients, the investors, whose risk tolerance is likely to be different from his and among each other’s. (The one exception is the so-called target-date funds or lifecycle funds, which are designed to suit the risk tolerance of the investors who hold them, though I believe that there is good reason to doubt that they do this at all well.) It is highly unlikely that any one of these investment bundles, on its own, will be right for you and your circumstances.

In my next three essays, the culmination of our explorations of basic investment concepts, we will see exactly how risk tolerance should be reckoned into the determination of a custom portfolio. We will see why, even if there are commonalities in expectations of return and risk, one person’s (or institution’s) portfolio is and ought to be different from another’s.

Yet my view of risk tolerance is pessimistic: We need to know it, and knowledge of it is close to unobtainable, and in any case uncertain and unverifiable. We can only contemplate the possibilities that investments hold out of reward and disappointment, and we must look to both at the same time that we look inside ourselves and judge our intrinsic emotional balance between the two.

Adam Jared Apt, CFA, is a financial advisor and the owner of Peabody River Asset Management, based in Cambridge, MA.

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