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The Stock Market: Beyond Risk Lies Uncertainty
Risk and uncertainty are manifestations of the same underlying force—randomness.
In fact, one of the most significant advances in understanding randomness
lies in distinguishing between the two concepts. Risk is randomness in
which events have measurable probabilities, wrote economist Frank Knight
in 1921 in Meaning of Risk and Uncertainty, a seminal treatment
on the topic.1
Probabilities may be attained either by deduction—using theoretical
models—or induction—using the observed frequency of events.
For instance, we can easily deduce the probabilities of the possible outcomes
of a game of dice. In a similar vein, economists deduce probability distributions
for stock market returns from theoretical models of investor behavior.
On the other hand, induction allows us to calculate probabilities from
past observations where theoretical models are unavailable, possibly because
of a dearth of knowledge about the underlying relation between cause and
effect. As an example, we can induce the probability of suffering a head
injury when riding a bicycle from observing how frequently it has happened
in the past. Similarly, economists estimate probability distributions
for stock market returns from the history of past returns. More intricate than risk is Knight's concept of uncertainty. Whereas
risk is quantifiable randomness, uncertainty isn't. It applies to situations
in which the world is not well-charted. First, our worldview might be
insufficient from the start, like Newtonian mechanics, which was proved
to be incomplete by Einstein's relativity. Second, the way the world works
itself might change, so that past observations offer little guidance for
the future. For instance, once bicyclists were encouraged to wear helmets,
the relation between riding the bicycle—the cause—and the
probability of suffering a head injury—the effect—changed.
You might simply think that the introduction of helmets would have reduced
the number of head injuries. Rather, the opposite happened. The number
of head injuries actually increased, possibly because helmet-wearing bikers
started riding in a more risky manner due to a false perception of safety.2
Paradoxically, the causality between riding the bicycle and suffering
a head injury changed because we started managing the observed
health risk based on the previously observed relation of cause and effect.
Typically, in situations of choice, risk and uncertainty both apply.
Many situations of choice are unprecedented, and uncertainty about the
underlying relation between cause and effect abounds. Given that risk is quantifiable, it is more accessible
to theoretical and empirical treatment than is uncertainty. It is thus
not surprising that academic literature on stock market randomness deals
exclusively with stock market risk. On the other hand, ignorance of uncertainty
may be hazardous to the investor's financial health, as the rise
and fall of Long-Term Capital Management illustrates. (See
more on LTCM's strategy.) Growth-Optimal Investment
For matters of illustrating risk, let us look at an (admittedly, rather
speculative) asset that returns 100 and negative 60 percent per year,
respectively, with equal probabilities. The statistical expected return
on the asset for the year is simply the average of the two possible returns,
20 percent. Thus, the expected value of a dollar invested in this asset
is $1.20 for a one-year investment horizon, $1.44 for a two-year horizon
and so on. For maximizing the statistical expected value of final wealth
(i.e., wealth at retirement), the investment appears worthwhile. For instance,
the expected value of that initial dollar investment 30 years down the
line is $237.38. Oddly enough, the investment in the risky asset is not so worthwhile.
The rate of capital growth of the $1 investment (with capital gains and
dividends reinvested) approaches negative 10.56 percent per year, which
is a far cry from the expected return on capital of 20 percent. On average,
the value of the portfolio increases by 100 percent one-half the time
and loses 60 percent the other half of the time. Over a two-year period,
then, the investor, on average, has only 80 percent of his portfolio value
remaining. This amounts to an average rate of capital growth of about
negative 10.56 percent per year. So, although the statistical value of
the expected final wealth increases with the investment horizon, the probability
of the investor actually enjoying this wealth decreases rapidly. Indeed,
if the investor is in for the long haul, he almost surely faces financial
ruin.3
(Any similarities between the risky asset and tech stocks are coincidental.) The puzzling difference between the expected return of 20 percent and
the expected growth rate of negative 10.56 percent results from the fact
that capital growth is multiplicative rather than additive due to compounding.
This is why a growth-optimal investment strategy is one that maximizes
the statistical expected value of the rate of capital growth, rather than
the expected value of final wealth (or, equivalently, expected return).
For any investment horizon, a growth-optimal investment strategy almost
surely leads to greater final wealth than any other investment strategy.
Put differently, the growth-optimal investment strategy almost surely
allows the investor to reach any targeted final wealth within the least
amount of time. On the other hand, an investor who maximizes expected
returns—rather than the expected rate of capital growth—almost
surely faces financial ruin in the long haul. Stock Market Risk The rate of capital growth of a buy-and-hold portfolio (that is, dividends
and capital gains reinvested) in the U.S. stock market is astounding.
One dollar invested at the end of 1925 in a buy-and-hold index portfolio
of large-capitalization stocks accumulated to $231.22 at the end of 2001,
even after adjusting for inflation. The implied average annual rate of
growth of this investment equals 7.43 percent. By comparison, the inflation-adjusted
average rate of growth of a buy-and-hold investment in long-term corporate
bonds averaged only about 2.63 percent. The corresponding numbers for
long-term government bonds, intermediate-term government bonds and Treasury
bills read 2.18, 2.22 and 0.73 percent.4
The superior performance of stocks relative to other standard types of
securities can be seen in other industrialized countries, too, as reported
in a study this year by economists Elroy Dimson, Paul Marsh and Mike Staunton.
As for risk, remember that a growth optimal investment strategy almost
surely offers an investor higher final wealth than any other investment
strategy. Thus, a growth optimal investment strategy already takes due
account of risk. In fact, the growth optimal investment strategy is a
survival-oriented concept of investment in risky assets. Jeremy Siegel
popularized the idea of the stock market as the growth optimal investment
strategy in his book Stocks for the Long Run, which was first published
in 1994. Not surprisingly, the fraction of families who own (directly
or indirectly) publicly traded stocks increased from 40.4 percent to 48.8
percent between 1995 and 1998.5
Stock Market Uncertainty
In trying to answer the question of whether historical data support the
hypothesis that a buy-and-hold stock market index portfolio is indeed
growth-optimal, it is tempting to look for an answer in stock markets
that exist at the present time. But there are pitfalls in this approach.
For instance, the aforementioned study by Dimson, Marsh and Staunton analyzes
only stock markets that have survived the vagaries of time and, not surprisingly,
are found in highly developed, wealthy countries. But there are also those
stock markets that have gone under. As economists Robert D. Arnott and
Peter L. Bernstein pointed out in a recent article, four stock markets
(China, Russia, Argentina and Egypt) suffered total capital loss—that
is, a return of negative 100 percent—during the 20th century. Note
that every one of these four countries was a significant economic or military
power at the time. Focusing on surviving stock markets and, in particular,
on the highly successful U.S. stock market distorts the picture—a
problem known as survivorship bias. As for the second question, the one about whether the world will behave
in the future as it did in the past, Warren Buffett presented last year
an interesting historical example of how our learning about the stock
market changes the way the market behaves. Buffett, the world's second-richest
man, distinguishes between periods of comparatively high and low stock
market valuation. In the early 1920s, stock market valuation was comparatively
low, as measured by the inflation-adjusted present value of future dividends.
The attractive valuation of stocks relative to bonds became a widely held
belief after Edgar Lawrence Smith published in 1924 a book on stock market
valuation, titled Common Stocks as Long Term Investments. Smith
argued that stocks not only offer dividends, but also capital appreciation
through retained earnings. The book, which was reviewed by John Maynard
Keynes in 1925, gave cause to an unprecedented stock market appreciation.
The inflation-adjusted average annual growth rate of a buy-and-hold investment
in large-company stocks established at the end of 1925 amounted to a staggering
32.13 percent at the end of 1928. On the other hand, over the next four
years, this portfolio depreciated at an average annual rate of 17.28 percent,
inflation-adjusted. Taken together, over the entire seven-year period,
the inflation-adjusted average annual growth rate of this portfolio came
to a meager 1.11 percent. Buy-and-hold portfolios in the allegedly unattractive
long-term corporate and government bonds, on the other hand, grew at inflation-adjusted
average annual rates of 10.18 and 9.83 percent, respectively! This proves
Buffett's point that, "What the few bought for the right reason
in 1925, the many bought for the wrong reason in 1929." We conclude
from this episode that learning about the stock market may feed back into
the market and, by changing the behavior of the market, render our "learning"
useless or, if we don't recognize the feedback effect, hazardous.
Let us now return to the present time. Figure 1 shows that, starting
in 1994, the ratio of stock market valuation to (nominal) GNP increased
sharply. Although the stock market valuation relative to GNP has declined
since its peak in the first quarter of 2000, this ratio is still at an
elevated level by historical standards. Is it possible that history is
repeating itself? Has learning about stock market risk led to complacency
about stock market uncertainty? The stock market outlook depends on whether the annual rates of growth
of a buy-and-hold stock market index portfolio are independent of past
rates of growth, or whether nature corrects past above-average growth
rates. To help understand this, imagine we flip a coin. We know that,
as we keep flipping the coin, the fraction of heads in the total number
of tosses converges to one-half. Yet, we also know that each toss is independent
of all others. Even if we observe 100 heads in a row, the probability
of tails coming up on the next toss is still one-half. The random process
dilutes deviations from the mean, rather than correcting them.6
On the other hand, sometimes outcomes are not independent of past realizations.
For instance, the expansion of fast-growing companies will slow inevitably
because no company can forever grow faster than the economy overall. (Any
similarities between mean-reversion of rates of growth and the demise
of once-admired tech companies are coincidental.)
Figure 2 exhibits the inflation-adjusted average growth rates of capital
invested in a buy-and-hold portfolio of large-company stocks. The black
line shows for any given date the average growth rate since year-end 1925.
The red line shows for any given date the average growth rate over the
prior 10 years. Barring changes in the size distribution of the economy's
corporate sector, one might expect the growth rate of capital invested
in large-company stocks to be proportional to the long-run growth rate
of the economy. Indeed, the average rate of growth for the portfolio established
in 1925 seems to approach a constant mean value. The 10-year growth-rate
averages, on the other hand, exhibit wild swings around this mean. If
the annual growth rates of capital in the stock market are random realizations,
the fact that the average rate of capital growth over the past 10 years
is above average contains no information for future rates of capital growth.
Future realizations will simply dilute the above-average rate of growth
of the past 10 years. However, if there is reversion to a constant mean,
then the rate of capital growth over the next 10 years of a buy-and-hold
portfolio invested in large-company stocks can be expected to be lower
than it was in the 1990s. In other words, if there is mean reversion in
stock market returns, the odds are stacked against the stock market. Academic
evidence on this question is not conclusive though--clearly, a case
of uncertainty. Conclusion Risk and uncertainty are two concepts that emanate from randomness. Neither
concept is fully understood. Although risk is quantifiable, uncertainty
is not. Rather, uncertainty arises from imperfect knowledge about the
way the world behaves. Most importantly, uncertainty relates to the questions
of how to deal with the unprecedented, and whether the world will behave
tomorrow the way it behaved in the past. For investors, not being able to distinguish between risk and uncertainty
is hazardous to their financial health. Although we have a fairly good
understanding of stock market risk, assessing stock market uncertainty
is incomparably harder. Ironically, the lower the level of risk, the more
aggressive are investors' bets, and the more vulnerable they are
to uncertainty. Clearly, a stock market valuation as elevated as it currently
is leaves much room for disappointment. Frank A. Schmid is a senior economist at the Federal Reserve Bank of St. Louis. Bill Bock provided research assistance.
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Life
is risky. The future is uncertain. These two phrases, I suppose, we
all have heard—and possibly used—many times. Although the words
risk and uncertainty seem to cross our lips easily, how well do we understand
the concepts? More specifically, are we aware of the implications
of risk and uncertainty for stock market investments? If the decline in
headline stock market indexes—such as the Dow Jones Industrial Average,
the Nasdaq Composite or the S&P 500—over the past two years has reminded
you of the possibility of steep declines, stay tuned.
Before
turning to an assessment of stock market risk and uncertainty, let us
briefly discuss the concept of growth-optimal investing, which is critical
to prospering in financial markets. To keep matters simple, we ignore
uncertainty. Rather, we go with the quantifiable, that is, risk. 
Stock
market uncertainty relates to imperfect information about how the world
behaves. First, how well do we understand the process that generated historical
stock market returns? More specifically, is the buy-and-hold investment
in a stock market index portfolio indeed the growth-optimal investment
strategy, as judged by past experience? Second, even if we had perfect
information about past processes, can we assume that the same relation
between cause and effect will apply in the future? In fact, the aforementioned
increase of head injuries in response to our (supposedly) improved understanding
of the causality between riding the bicycle and suffering a head injury
suggests that this might not be the case.
