Capitalism and Society
Volume 1, Issue 2 2006 Article 5
Investing
in
the Unknown and Unknowable
Richard Zeckhauser - Harvard University
Copyright (c) 2006 The Berkeley Electronic Press. All rights reserved
Abstract
From David Ricardo making a fortune buying British government bonds on the eve of the Battle of Waterloo to Warren Buffett selling insurance to the California earthquake authority, the wisest investors have earned extraordinary returns by investing in the unknown and hte unknowable (UU). But they have done so on a reasoned, sensible basis. This easay explains some of the central principles that such investors employ. It starts by discussing "ignorance," a widespread situation in the real world of investing, where even the possible states of the world are not known. Traditional finance theory does not apply in UU situations.
Strategic thinking, deducing what other investors might know or not, and assessing whether they might be deterred from investing, for example due to fiduciary requirements, frequently point the way to profitability. Most big investment payouts come when money is combined with complementary skills, such as knowing how to develop real estate or new technologies. Those who lack these skills can look for "sidecar" investments that allow them to put their money alongside that of people they know to be both capable and honest. The reader is asked to consider a number of usch investments.
Central concepts in decision analysis, game theory, and behavioral decision are deployed along-side real investment decisions to unearch successful investment strategies. There strategies are distiled into eight investment maxims. Learning to invest more wisely in a UU world may be the most promising way to significantly bolster your properity.
KEYWORDS: investing, unknown, unknowable, sidecar investment, fattailed distribution, buffett, Kelly Criterion, asymmetric information.
David Ricardo made a fortune buying bonds from
the British government four days in advance of the Battle of Waterloo. He was
not a military analyst, and even if he were, he had no basis to compute the
odds of Napoleon’s defeat or victory, or hard-to-identify ambiguous outcomes.
Thus, he was investing in the unknown and the unknowable. Still, he knew that
competition was thin, that the seller was eager, and that his windfall pounds
should Napoleon lose would be worth much more than the pounds he’d lose should
Napoleon win. Ricardo knew a good bet when he saw it.1
This essay discusses how
to identify good investments when the level of uncertainty is well beyond that
considered in traditional models of finance. Many of the investments considered
here are one-time only, implying that past data will be a poor guide. In
addition, the essay will highlight investments, such as real estate
development, that require complementary skills. Most readers will not have such
skills, but many will know others who do. When possible, it is often wise to
make investments alongside them.
Though investments are the
ultimate interest, the focus of the analysis is how to deal with the unknown
and unknowable, hereafter abbreviated UU.
Hence, I will sometimes discuss salient problems outside of finance, such
as terrorist attacks, which are also unknown and unknowable.
This essay takes no
derivatives, and runs no regressions.2 In short, it eschews the
normal tools of my profession. It represents a blend of insights derived from
reading academic works and from trying to teach their insights to others, and
from lessons learned from direct and at-a-distance experiences with a number of
successful investors in the UU world. To reassure my academic audience, I use
footnotes where possible, though many refer to accessible internet articles in
preference to journals and books. Throughout this essay, you will find
speculations and maxims, as seems called for by the topic. They will be labeled
in sequence.
This informal approach
seems appropriate given our present understanding of the topic. Initial beliefs
about this topic are highly uncertain, or as statisticians would phrase it:
“Prior distributions are diffuse.” Given that, the judicious use of
illustrations, and prudent attempts to provide taxonomies and sort tea leaves,
can substantially hone our beliefs, that is, tighten our future predictions.
Part I of this essay talks
about risk, uncertainty, and ignorance, the last carrying us beyond traditional
discussions. Part II looks at behavioral economics, the tendency for humans to
deviate in systematic ways from rational decision, particularly when
probabilities are involved, as they always are with investments. Behavioral
economics pervades the UU world. Part III addresses the role of skilled
mathematical types now so prevalent in finance. It imparts a general lesson: If
super-talented people will be your competitors in an investment arena, perhaps
it is best not to invest. Its second half discusses a dispute between math
types on money management, namely how much of your money to invest when you do
have an edge. Part IV details when to
invest when you can make more out of an investment, but there is a better
informed person on the other side of the transaction. Part V tells a Buffett
tale, and draws appropriate inferences. Part VI concludes.
I. RISK, UNCERTAINTY AND IGNORANCE
Escalating challenges to
effective investing. The essence of effective
investment is to select assets that will fare well when future states of the
world become known. When the probabilities of future states of assets are
known, as the efficient markets hypothesis posits, wise investing involves
solving a sophisticated optimization problem. Of course, such probabilities are
often unknown, banishing us from the world of the capital asset pricing model
(CAPM), and thrusting us into the world of uncertainty.3
Were the financial world
predominantly one of mere uncertainty, the greatest financial successes would
come to those individuals best able to assess probabilities. That skill, often
claimed as the domain of Bayesian decision theory, would swamp sophisticated
optimization as the promoter of substantial returns.
The real world of investing
often ratchets the level of non-knowledge into still another dimension, where
even the identity and nature of possible future states are not known. This is
the world of ignorance. In it, there is no way that one can sensibly assign
probabilities to the unknown states of the world. Just as traditional finance
theory hits the wall when it encounters uncertainty, modern decision theory
hits the wall when addressing the world of ignorance. I shall employ the
acronym UU to refer to situations where both the identity of possible future
states of the world as well as their probabilities are unknown and unknowable.
Table 1 outlines the three escalating categories; entries are explained
throughout the paper.
This essay has both dreary
and positive conclusions about investing in a UU world. The first dreary
conclusion is that unknowable situations are widespread and inevitable.
Consider the consequences for financial markets of global warming, future terrorist
activities, or the most promising future technologies. These outcomes are as
unknowable today as were the 1997 Asian meltdown, the 9/11 attacks, or the
NASDAQ soar and swoon at the end of the century, shortly before they were
experienced.
These were all aggregate
unknowables, affecting a broad swath of investors. But many unknowables are
idiosyncratic or personal, affecting only individuals or
miles to the west of the city, will they
come? Will the Vietnamese government let me sell my insurance product on a
widespread basis? Will my friend’s new software program capture the public
fancy, or if not might it succeed in a completely different application? Such
idiosyncratic UU situations, I argue below, present the greatest potential for
significant excess investment returns.
The second dreary
conclusion is that most investors – whose training, if any, fits a world where
states and probabilities are assumed known – have little idea of how to deal
with the unknowable. When they recognize its presence, they tend to steer
clear, often to protect themselves from sniping by others. But for all but the
simplest investments, entanglement is inevitable – and when investors do get
entangled they tend to make significant errors.
The first positive
conclusion is that unknowable situations have been and will be associated with
remarkably powerful investment returns. The second positive conclusion is that
there are systematic ways to think about unknowable situations. If these ways
are followed, they can provide a path to extraordinary expected investment
returns. To be sure, some substantial losses are inevitable, and some will be
blameworthy after the fact. But the net expected results, even after allowing
for risk aversion, will be strongly positive.
Do not read on, however,
if blame aversion is a prime concern: The world of UU is not for you. Consider
this analogy. If in an unknowable world none of your bridges fall down, you are
building them too strong. Similarly, if in an unknowable world none of your
investment looks foolish after the fact, you are staying too far away from the
unknowable.
Warren Buffett, a master
at investing in the unknowable, and therefore a featured player in this essay,
is fond of saying that playing contract bridge is the best training for
business. Bridge requires a continual effort to assess probabilities in at best
marginally knowable situations, and players need to make hundreds of decisions
in a single session, often balancing expected gains and losses. But players
must also continually make peace with good decisions that lead to bad outcomes,
both one’s own decisions and those of a partner. Just this peacemaking skill is
required if one is to invest wisely in an unknowable world.
The nature of unknowable
events. Many of the events that we classify as
unknowable arrive in an unanticipated thunderclap, giving us little or no time
to anticipate or prepare. But once they happen, they do not appear that
strange. The human mind has an incredible ability to find a rationalization for
why it should have been able to conjecture the terror attack of 9/11; or the
Asian tsunamis of 1997 and 2005, respectively caused by currency collapse and
underwater earthquake. This propensity to incorporate hindsight into our
memories – and to do so particularly our ability to anticipate extreme events in the future. We learn
insufficiently from our misestimates and mistaken decisions.
Other unknowable events
occur over a period of time, as did the collapse of the Soviet Union. Consider
most stock market swings. Starting in January 1996, the NASDAQ rose five-fold
in four years. Then it reversed field and fell by two thirds in three years.
Such developments are hardly thunderclaps. They are more like blowing up a
balloon and then dribbling out the air. In retrospect, these remarkable swings
have lost the flavor of an unknowable event, even though financial markets are
not supposed to work that way. If securities prices at any moment incorporate
all relevant information, a property that is usually posited, long-term
movements in one direction are hardly possible, since strong runs of
unanticipated good news or bad news will be exceedingly rare. Similarly, the
AIDS scourge now seems familiar territory, though 25 years ago – when there had
been only 31 cumulative deaths in the U.S. from AIDS – no one would have
predicted a world-wide epidemic killing tens of millions and vastly disrupting
the economies of many poor nations.
Are UU events to be
feared? Warren Buffett (1996) once remarked: “It is essential to remember that
virtually all surprises are unpleasant.”
Most salient UU events seem to
fall into the left tail of unfortunate occurrences. This may be more a matter
of perception than reality. Often an upside unknowable event, say the
diminution of terror attacks or recovery from a dread disease, is difficult to
recognize. An attack on any single day was not likely anyway, and the patient
still feels lousy on the road to recovery. Thus, the news just dribbles in, as
in a financial market upswing. B.F. Skinner, the great behavioral psychologist,
taught us that behavior conditioned by variable interval reinforcement – engage
in the behavior and from time-to-time the system will be primed to give you a
payoff – was the most difficult to extinguish. Subjects could never be sure
that another reward would not be forthcoming. Similarly, it is hard to discern
when a string of inconsistently spaced episodic events has concluded. If the
events are unpleasant, it is not clear when to celebrate their end.
Let us focus for the
moment on thunderclap events. They would not get this title unless they
involved something out of the ordinary, either good or bad. Casual empiricism –
judged by looking at local, national and international headlines – suggests
that thunderclap events are disproportionately adverse. Unlike in the old
television show, The Millionaire,
people do not knock on your door to give you a boatload of money, and in Iraq
terror attacks outnumber terrorist arrests manifold.
The financial arena may be
one place with an apparently good ratio of upside to downside UU events,
particularly if we include events that are drifts and not thunderclaps. By the
end of 2004, there were 2.5 million millionaires in the United States,
excluding housingwealth.
http://money.cnn.com/2005/06/09/news/world_wealth/ Many of these individuals, no doubt, experienced upside UU events.
Some events, such as the sustained boom in housing prices, were experienced by
many, but many upside events probably only affected the individual and perhaps
a few others; such events include an unexpected lucrative job, or having a
business concept take a surprisingly prosperous turn, or having a low-value
real estate holding explode in value, etc.
We hear about the lottery
winner -- the big pot, the thunderclap, and the gain for one individual makes
it newsworthy. In contrast, the tens of thousands of UU events that created
thousands of new real estate millionaires are mostly reported in dry aggregate
statistics. Moreover, contrary to the ads in the back of magazines, there is
usually not a good way to follow these “lucky folks,” since some complementary
skill or knowledge is likely to be required, not merely money and a wise choice
of an investment. Thus, many favorable UU financial events are likely to go
unchronicled.
While still in this
Pollyannish frame, it is worth noting the miracles of percentage symmetry given
extreme events. Posit that financial prices move in some symmetric fashion.
Given that negative prices are not possible, such changes must be in percentage
rather than absolute terms.4
We will not notice any difference
between percentage and absolute if changes are small relative to the mean.
Thus, if a price of 100 goes up or down by an average of 3 each year, or up by
a ratio of 103/100 or down by 100/103 hardly matters. But change that 3 to a
50, and the percentage symmetry helps a great deal. The price becomes
100(150/100) or 100(100/150)), which has an average of 117. If prices are
anything close to percentage symmetric, as many believe they are, then big
swings are both enemy and friend: enemy because they impose big risks, friend
because they offer substantial positive expected value.
Many millionaires have
made investments that multiplied their money 10- fold, and some 100-fold. The
symmetric geometric model would expect events that cut one’s stake to 1/10th
or 1/100th of its initial value to be equally likely. The
opportunity to get a 10 or 100 multiple on your investment as often as you lose
virtually all of it is tremendously attractive.
There is, of course, no
reason why investments must yield symmetric geometric returns. But it would be
surprising not to see significant expected excess returns to investments that
have three characteristics addressed in this essay: (1) UU underlying features,
(2) complementary capabilities are required
to
undertake them, so the investments are not
available to the general market, and
(3) it is unlikely that a party on the other
side of the transaction is better informed. That is, UU may well work for you,
if you can identify general characteristics of when such investments are desirable,
and when not.
These very attractive
three-pronged investments will not come along everyday. And when they do, they
are unlikely to scale up as much as the
investor would like, unlike an investment in an underpriced NYSE stock,
which scales nicely, at least over the range for most individual investors.
Thus, the UU- sensitive investor should be constantly on the lookout for new
opportunities. That is why Warren Buffett trolls for new businesses to buy in
each Berkshire- Hathaway annual report, and why most wealthy private investors
are constantly looking for new instruments or new deals.
Uniqueness. Many UU situations deserve a third U, for unique. If they do,
arbitrageurs – who like to have considerable past experience to guide them –
will steer clear. So too will anybody who would be severely penalized for a
poor decision after the fact. An absence of competition from sophisticated and
well- monied others spells the opportunity to buy underpriced securities.
Most great investors, from
David Ricardo to Warren Buffett, have made most of their fortunes by betting on
UUU situations. Ricardo allegedly made 1 million pounds (over $50 million
today) – roughly half of his fortune at death – on his Waterloo bonds.5
Buffett has made dozens of equivalent investments. Though he is best known for
the Nebraska Furniture Mart and See’s Candies, or for long-term investments in
companies like the Washington Post and Coca Cola, insurance has been Berkshire
Hathaway’s firehose of wealth over the years. And insurance often requires UUU
thinking. A whole section below discusses Buffett’s success with what many
experts saw as a UUU insurance situation, so they steered clear; but he saw it
as offering excess premium relative to risk, so he took it all.
Speculation 1: UUU investments – unknown,
unknowable and unique – drive off speculators, which creates the potential for
an attractive low price.
Some UU situations that
appear to be unique are not, and thus fall into categories that lend themselves
to traditional speculation. Corporate
takeover bids are such situations. When one company makes a bid for another, it is often impossible to determine
what is going on or what will happen, suggesting uniqueness. But since dozens
of such situations have been seen over the years, speculators are willing to
take positions in them. From the standpoint of investment, uniqueness is lost,
just as the uniqueness of each child matters not to those who manufacture
sneakers.
Weird Causes and Fat Tails. The returns to UUU investments can be extreme. We are all familiar
with the Bell Curve (or Normal Distribution), which nicely describes the number
of flips of a fair coin that will come up heads in a large number of trials.
But such a mechanical and controlled problem is extremely rare. Heights are
frequently described as falling on a Bell Curve. But in fact there are many too
many people who are extremely tall or extremely short, due say to glandular
disturbances or genetic abnormalities. The standard model often does not apply
to observations in the tails. So too with most disturbances to investments.
Whatever the explanation for the October 1987 crash, it was not due to the
usual factors that are used to explain market movements.6
More generally, movements
in financial markets and of investments in general appear to have much thicker
tails than would be predicted by Brownian motion, the instantaneous source of
Bell Curve outcomes. That may be because the fundamental underlying factors
produce thicker tails, or because there are rarely occurring anomalous or weird
causes that produce extreme results, or both. The UU and UUU models would give
great credence to the latter explanation, though both could apply.7
Complementary skills and UU
investments. A great percentage of UU investments,
and a greater percentage of those that are UUU, provide great returns to a complementary
skill. For example, many of America’s great fortunes in recent years have come
from real estate. These returns came to people who knew where to build, and
what and how. Real estate developers earn vast amounts on their capital because they have
complementary skills. Venture capitalists can secure extraordinary returns on
their own monies, and charge impressive fees to their investors, because early
stage companies need their skills and their connections. In short, the return
to these investments comes from the combination of scarce skills and wise
selection of companies for investment. High tech pioneers – Bill Gates is an
extreme example – get even better multiples on their investment dollars as a
complement to their vision and scientific insight.8
Alas, few of us possess
the skills to be a real estate developer, venture capitalist or high tech pioneer.
But how about becoming a star of ordinary stock investment? For such efforts an
ideal complementary skill is unusual judgment. Those who can sensibly determine
when to plunge into and when to refrain from UUU investments gain a substantial
edge, since mispricing is likely to be severe. Bill Miller, the famed manager
of the Legg Mason Value Fund, had a unique record of beating the S&P; his
string through December 2005 was 15 years in a row. In October 2004 he spoke at
Harvard University, and explained in detail why he made major purchases of
Google at its public offering, surely a UUU situation given the nature of the
company and the fact that it was offered through a Dutch auction.9
Virtually all in the audience were impressed that he made this decision -- the
stock came out at $85 in August that year and had run up to $140. But Miller
recognized that explaining past successes is not a challenge. He went on to
proclaim Google a great investment for the future. How right he was. Google was
selling at $380 in September 2006, when this essay was completed. Alas, 2006
was not kind to Miller. By September, his Value Fund was 12% behind the S&P
for the year. Only time will tell whether Miller has lost his touch or is
merely in a slump.
Warren Buffett’s unusual
judgment operates with more prosaic
companies, such as oil producers and soft drink firms. He is simply a
genius at everyday tasks, such as judging management capability or forecasting
company progress. He drains much of the unknowable in judging a company’s
future. But he has other advantages. A number of Buffett’s investments have
come to him because companies sought him out, asking him to make an investment
and also to serve on their board, valuing his discretion, his savvy, and his
reputation for rectitude – that is, his complementary skills, not merely his
money. And when he is called on for such reasons, he often gets a discounted price. Those like Miller
and Buffet, who can leverage complementary
skills in stock market investment, will be in a privileged position of limited
competition. But that will accomplish little if they do not show courage and
make big purchases where they expect high payoffs. But the lesson for regular
mortals is not to imitate Warren Buffett or Bill Miller; that makes no more
sense than trying to play tennis like Roger Federer. Each of them has an
inimitable skill. If you lack Buffett-Miller capabilities, you will get chewed
up as a bold stock picker.
Note, by the way, the
generosity with which great investors with complementary skills explain their
successes – Buffett in his annual reports, Miller at Harvard, and any number of
venture capitalists who come to lecture to MBAs. These master investors need
not worry about the competition, since few others possess the complementary
skills for their types of investments. Few UU investment successes come from
catching a secret, such as the whispered hint of “plastics” in the movie The Graduate. Mayer Amschel Rothschild
had five sons who were bright, disciplined, loyal and willing to disperse.
These were the complementary skills. The terrific investments in a UU world –
and the Rothschild fortune – followed.
Before presenting a maxim
about complementary skills, I present you with a decision problem. You have
been asked to join the Business Advisory Board of a company named Tengion.
Tengion was founded in 2003 to develop and commercialize a medical
breakthrough: “developing new human tissues and organs (neo-tissues and
neo-organs)
that are derived from a patient’s own cells…[this technology] harnesses the
body’s ability to regenerate, and it has the potential to allow adults and
children with organ failure to have functioning organs built from their own (autologous)
tissues.” http://www.tengion.com/
This is assuredly a UU
situation, doubly so for you, since until now you had never heard the term
neo-organ. A principal advantage of joining is that you would be able to invest
a reasonable sum on the same basis as the firm’s insiders and venture
capitalists. Would you choose to do so?
I faced this decision
problem because I had worked successfully with Tengion’s president on another
company many years earlier. I was delighted with the UU flavor of the
situation, and chose to join and invest because I would be doing so on the same
terms as sophisticated venture capital (VC) firms with track records and
expertise in relevant biotech areas. This was an investment from which
virtually everyone else would be excluded. In addition, it would benefit from
the complementary skills of the VCs.
Sidecar investments. Such undertakings are “sidecar investments”; the investor rides
along in a sidecar pulled by a powerful motorcycle. The more the investor is
distinctively positioned to have confidence in the driver’s integrity and his
motorcycle’s capabilities, the more attractive the investment, since its price will
be lower due to limited competition. Perhaps the premier sidecar investment
ever available to the ordinary investor was Berkshire Hathaway, many decades
back. One could have invested alongside Warren Buffett, and had him take a
ridiculously low compensation for his services. (In recent years, he has been
paid
$100,000, with no bonus or options.) But in
1960 who had heard of Warren Buffett, or knew that he would be such a
spectacular and poorly compensated investor? Someone who knew Buffett and
recognized his remarkable capabilities back then was in a privileged UU situation.
Maxim A: Individuals with complementary
skills enjoy great positive excess returns from UU investments. Make a sidecar
investment alongside them when given the opportunity.
Do you have the courage to
apply this maxim? It is January 2006 and you, a Western investor, are deciding
whether to invest in Gazprom, the predominantly government-owned Russian
natural gas giant in January 2006. Russia is attempting to attract
institutional investment from the West; the stock is sold as an ADR, and is
soon to be listed on the OTC exchange; the company is fiercely profitable, and
it is selling gas at a small fraction of the world price. On the upside, it is
generally known that large numbers of the Russian elite are investors, and here
and there it is raising its price dramatically. On the downside, Gazprom is
being employed as an instrument of Russian government policy, e.g., gas is sold
at a highly subsidized price to Belarus, because of its sympathetic government,
yet the Ukraine is being threatened with more than a four-fold increase in
price, in part because its government is hostile to Moscow. And the company is
bloated and terribly managed. Finally, experiences, such as those with Yukos
Oil, make it clear that the government is powerful, erratic, and ruthless.
This is clearly a
situation of ignorance, or UU. The future states of the world are simply not
known. Will the current government stay in power? Will it make Gazprom its
flagship for garnering Western investment? If so, will it streamline its operations?
Is it using foreign policy concerns as a device mainly to raise prices, a
strong positive, and is it on a path to raise prices across the board? Will it
complete its proposed pipelines to Europe?
What questions haven’t you thought of, whose answers could dramatically
affect your payout? Of course, you should also determine whether Western
investors have distinct disadvantages as Gazprom shareholders, such as unique
taxes, secondary voting status, etc. Finally, if you determine the investment
is favorable given present circumstances, you should ask how quickly Russia
could change conditions against outsiders, and whether you will be alert and
get out if change begins.
You could never learn
about the unknowables sufficiently well to do traditional due diligence on a
Gazprom investment. The principal arguments for going ahead would be that
Speculation 1 and Maxim A apply. If you could comfortably determine that the
Russian elite was investing on its own volition, and that foreigners would not
be discriminated against, or at least not quickly, this would make a sensible
sidecar investment.10
II. BEHAVIORAL ECONOMICS AND DECISION TRAPS
Behavioral decision has shaken the fields of
economics and finance in recent decades. Basically, this work shows in area
after area that individuals systematically deviate from making decisions in a
manner that would be admired by Jimmie Savage (1954) and Howard Raiffa (1968),
pioneers of the rational decision paradigm. As one illustration, such deviators
could be turned into money pumps: They would pay to pick gamble B over gamble
A. Then with A reframed as A’, but not changed in its fundamentals, they would
pay to pick A over B.
That is hardly the path to
prudent investment, but alas behavioral decision has strong descriptive
validity. Behavioral decision has important implications for investing in UU situations. When
considering our own behavior, we must be extremely careful not to fall prey to
the biases and decision traps it chronicles. Almost by definition, UU situations
are those where our experience is likely to be limited, where we will not
encounter situations similar to other situations that have helped us hone our intuition.
Virtually all of us fall
into important decision traps when dealing with the unknowable. This section
discusses two, overconfidence and recollection bias, and then gives major
attention to a third, misweighting differences in probabilities and payoffs.
But there are dozens of decision traps, and some will appear later in this
essay. The Nobel Prize winning work of Daniel Kahneman and Amos Tversky (the
latter was warmly cited, but died too soon to win), 11 and the
delightful and insightful Poor Charlie’s
Almanack, written by Charles Munger (Warren Buffett’s partner) respectively
provide academic and finance-oriented discussions of such traps.
There are at least three
major objections to behavioral economics: First, in competitive markets, the
anomalies it describes will be arbitraged away. Second, the anomalies only
appear in carefully crafted situations; they are much like optical illusions,
intriguing but rarely affecting everyday vision. Third, they describe the objection
is tangential to this discussion; competitive markets and arbitrage are not
present in many UU situations, and in particular not the ones that interest us.
The second objection is relatively unimportant because, in essence, UU
situations are those where optical illusions rule the world. A UU world is not
unlike a Fun House. Objection three I take up seriously below; this essay is
designed to help people behave more rationally when they invest.
Let us first
look at the biases.
Overconfidence. When individuals are assessing quantities about which they know very
little, they are much too confident of their knowledge (Alpert and Raiffa,
1982). Appendix A offers you a chance to test your capabilities in this
regard. For each of eight unknown
quantities, such as the area of Finland, you are asked to provide your median
estimate, then your 25th and 75th percentile estimates
(i.e., it is one quarter likely the true value will be more extreme than either
of the two), and then your 1st and 99th percentiles, what
are referred to as surprise points. In theory, an individual should have
estimates outside her surprise points about 2% of the time. In fact, even if
warned about overconfidence, individuals are surprised about 35% of the time.12
Quite simply, individuals think they know much more about unknowable quantities
than they do.
Speculation 2: Individuals who are
overconfident of their knowledge will fall
prey to poor investments in the UU world. Indeed, they are the green
plants in the elaborate ecosystem of finance where there are few lions, like
Bill Miller and Warren Buffett; many gazelles, like you and me; and vast acres
of grass ultimately nourishing us all.
Recollection bias. A first lesson in dealing with UU situations is to know thyself. One
good way to do this is to review successes and failures in past decisions.
However, since people do not have a long track record, they naturally turn to
hypotheticals from the past: Would I have judged the event that actually
occurred to be likely? Would I have made that good investment and steered clear
of the other bad one? Would I have sold out of NASDAQ stocks near New Year
2001? Alas, human beings do not do well with such questions. They are subject
to substantial recollection bias.13
Judging by articles in the
New York Times leading up to
9/11/2001, there was a clear UUU event. But that is not what respondents told
us one to three years later. They were
asked to compare their present assessments of the likelihood of a massive
terrorist attack with what they estimated that likelihood to be on September 1,
2001. Of more than 300 Harvard Law and Kennedy School students surveyed, 31%
rated the risk as now lower, and 26% rated the risk as the same as they had
perceived the 9/11 risk before the event.14 We can hardly be
confident that investors will be capable of judging how they would have
assessed UU risks that occurred in the past.
Misweighting probabilities
and preferences. The two critical components of
decision problems are payoffs and probabilities. Effective decision requires
that both be carefully calibrated. Not surprisingly, Prospect Theory, the most
important single contribution to behavioral decision theory to date, finds that
individuals’ responses to payoffs and probabilities are far from rational.15
To my knowledge, there is no tally of which contributes more to the loss of
expected utility from the rational norm. (Some strong supporters of behavioral
decision theory, however, think it is our norms that are misguided, and that
the way the brain naturally perceives outcomes, not the prescriptions of
decision theorists and economists, should be the guideline.)
Whether drawing from
Prospect Theory or observation, it seems clear that individuals draw
insufficient distinctions among small probabilities. Consider the following
experiment, in which an individual is asked to pick A or B.
Lottery Choice: Payoffs Versus
Probabilities
|
|
Payoff
|
Probability
|
|
A
|
$2000
|
0.01
|
|
B
|
$1000
|
0.025
|
A rational, risk averse
individual should opt for B, since it offers a higher expected value – $25
versus $20 – and less risk. Yet past experiments have shown that many individuals choose A, since
in accordance with Prospect Theory they do not distinguish sufficiently between
two low probability events. We speculate further that if we used named
contingencies – for example, the Astros or the Blue Jays win the World Series –
alongside their probabilities, the frequency of preference for A would
increase. The contingencies would be selected, of course, so that their likelihood
of occurrence, as indicated by odds in Las Vegas, would match those in the
example above.
This hypothetical
experiment establishes a baseline for another one that involves UU events. This
time the prizes are based on events that are as close to the spectrum of UU
events as possible, subject to the limitation that they must be named.16
Thus, a contingency might be that a 10,000-ton asteroid passed within 50,000
miles of Earth within the past decade, or that more than a million mammals
crossed the border from Tanzania to Kenya last year. To begin our experiment,
we ask a random sample of people to guess the likelihood of these
contingencies. We then alter the asteroid distance or the number of animals in
the question until the median answer is 0.03. Thus, if 50,000 miles got a
median answer of 0.05, we would adjust to 40,000 miles, etc.
We now ask a new group of
individuals to choose between C and D, assuming that we have calibrated the
asteroid and mammal question to get to
0.03.
Lottery Choice: Payoffs Versus
Probability or UU Event
|
|
Payoff
|
Required
contingency
|
|
C
|
$2000
|
Draw a 17 from
an urn with balls numbered 1 to 100
|
|
D
|
$1000
|
10,000-ton
asteroid passed within 40,000 miles of Earth
|
Lotteries C and D should
yield their prizes with estimated probabilities of 1% and 3% respectively.
Still, we suspect that many more people would pick C over D than picked A over
B, and that this would be true for the animal
movement contingency as well.17
A more elaborated version
of this problem would offer prizes based on alternative UU contingencies coming
to pass. For example, we might recalibrate the mammal-crossing problem to get a
median response of 0.01. We would then have:
Lottery Choice: Payoffs Versus
UU Events
|
|
Payoff
|
Required contingency
|
|
E
|
$2000
|
Calibrated large number of animals crossed the Tanzania- Kenya border
|
|
F
|
$1000
|
10,000
ton-asteroid passed within 40,000 miles of Earth
|
Here the values have been
scaled so the median response is three times higher for the asteroid event than
the animal crossing. We would conjecture again that E would be chosen
frequently.18 People do not like to rely on the occurrence of UU
events, and choices based on distinguishing among their probabilities would be
an unnatural act.
Daniel Ellsberg (1961)
alerted us to ambiguity aversion long before he created a UU event by
publishing the Pentagon papers. In an actual experiment, he showed, in effect,
that individuals preferred to win a prize if a standard coin flip came up
heads, rather than to win that prize by choosing either heads or tails on the
flip of a mangled coin whose outcome was difficult to predict.19
Such ambiguity aversion may be a plausible heuristic response to general
decisions under uncertainty, since so often there is a better-informed person
on the other side – such as someone selling a difficult-to-assess asset.20
Whatever the explanation, ambiguity aversion has the potential to exert a
powerful effect. Extending Ellsberg one step further, it would seem that the
more ambiguous the contingencies, the greater the aversion. If so, UU
investments will drive away all but the most self-directed and rational
thinking investors. Thus, Speculation 1 is reinforced.
III. MATH WHIZZES IN FINANCE AND CASH MANAGEMENT
The major fortunes in finance, I would
speculate, have been made by people who are effective in dealing with the
unknown and unknowable. This will probably be truer still in the future. Given
the influx of educated professionals into finance, those who make their living
speculating and trading in traditional markets are increasingly up against
others who are tremendously bright and tremendously well-informed.21
By contrast, those who
undertake prudent speculations in the unknown will be amply rewarded. Such
speculations may include ventures into uncharted areas, where the finance
professionals have yet to run their regressions, or may take completely new
paths into already well-traveled regions.22 It used to be said that
if your shoeshine boy gave you stock tips it was time to get out of the market.
With shoeshine boys virtually gone and finance Ph.D.’s plentiful, the new
wisdom might be:
When your math whiz finance Ph.D. tells you
that he and his peers have been hired to work in the XYZ field, the spectacular
returns in XYZ field have probably vanished forever.
Similarly, the more difficult a field is to
investigate, the greater will be the unknown and unknowables associated with
it, and the greater the expected profits to those who deal sensibly with them.
Unknownables can’t be transmuted into sensible guesses -- but one can take
one’s positions and array one’s claims so that unknowns and unknowables are
mostly allies, not nemeses. And one can train to avoid one’s own behavioral
decision tendencies, and to capitalize on those of others.
Assume that an investor is willing to
invest where he has an edge in UU situations. How much capital should then be
placed into each opportunity? This problem is far from the usual portfolio problem. It is afflicted
with ignorance, and decisions must be made in sequential fashion. Math whizzes
have discussed this problem in a literature little known to economists, but
frequently discussed among gamblers and mathematicians. The most famous
contribution is an article published 50 years ago by J.L. Kelly, an AT&T
scientist. His basic formula, which is closely related to Claude Shannon’s
information theory, tells you how much to bet on each gamble as a function of
your bankroll, with the probability of winning and the odds as the two
parameters. Perhaps surprisingly, the array of future investment opportunities
does not matter.
Kelly’s Criterion, as it
is called, is to invest an amount equal to W – (1- W)/R, where W is your
probability of winning, and R is the ratio of the amount you win when you win
to the amount you lose when you lose. Thus, if you were 60% likely to win an
even money bet, you would invest .6 – (1-.6)/1 = .2 or 20% of your capital.
It can be shown that given
sufficient time, the value given by any other investment strategy will
eventually be overtaken in value by following the Kelly Criterion, which
maximizes the geometric growth rate of the portfolio. That might seem to be definitive. But even in the
mathematical realm of optimal dynamic investment strategies, assuming that all
odds and probabilities are known, we encounter a UU situation.
Paul Samuelson, writing in
a playful mood, produced an article attacking the Kelly Criterion as a guide
for practice. His article uses solely one-syllable words. His abstract
observes: “He who acts in N plays to make his mean log of wealth as big as it
can be made will, with odds that go to one as N soars, beat me who acts to meet
my own tastes for risk.”23
Samuelson correctly prescribes that
in favorable-odds situations, whether repeated or not, the optimal
amount for an individual who maximizes his expected utility to invest will
depend on his utility function. To promote your intuition, consider a polar
case. A risk-neutral investor should invest his total wealth
whenever he confronts a favorable-odds situation, as opposed to the “magic
fraction” proposed by Kelly. Going all in, to use poker terminology, will
maximize his expected total wealth, hence his expected utility, for any finite
number of periods.24 In short, Samuelson shows that the Kelly
Criterion, though mathematically correct, should not guide an
investor’s actions, since it ignores the
structure of preferences, whether risk neutral or risk averse.25
Accounting for preferences, it turns out that
the Kelly Criterion leads to precisely the right investment proportions if
one’s utility function is logarithmic, but it is too conservative for less
risk-averse utility functions, and vice versa.
With logarithmic utility, one will just take an even money bet that
either multiplies one’s wealth by 1+x or by 1/(1+x), for any x. Thus, one would
take an even money bet to double or halve one’s wealth.
I lack both the space and
capability to straighten out the sequential investment problem. But I should
make a few observations to point out that even if the Kelly Criterion were
correct, the formulation it employs does not capture most real world investment
opportunities: (1) Most UU investments are illiquid for a significant period,
often of unknown length. Monies invested today will not be available for
reinvestment until they become liquid. (2) Markets charge enormous premiums to
cash out illiquid assets.26 (3) Models of optimal
sequential investment strategies tend to assume away the most important
real- world challenges to such strategies, such as uncertain lock-in periods.
(4) There are substantial disagreements in the literature even about “toy
problems,” such as those with immediate resolution of known-probability
investments. The overall conclusion is that: (5) Money management is a
challenging task in UU problems. It afflicts even those with a substantial edge
when making such investments. And when the unknowable happens, as it did with
the air- pocket plunge in the 1987 stock market or the 1997 Asian crisis,
unforeseen short-term money-management problems – e.g., transferring monies
across markets in time to beat margin calls – tend to emerge. These five points
imply that even if it were clear how one should invest in a string of favorable
gambles each of which is resolved instantaneously, that would help us little in
the real world of UU investing, which presents a much more difficult task.
Though I have quibbled
about the Kelly Criterion, it makes a simple, central point that is missed in
virtually all investment advice. Most such advice focuses on efficient or near efficient markets, implying that one
will not have a great edge in any investment. In contrast, the real world
presents some ordinary investments, some attractive investments, and some very
attractive investments. Clearly it makes sense to invest more in the more
attractive investments. This leads to a maxim on investment advantage:
Maxim B: The greater is your expected return
on an investment, that is the larger is your advantage, the greater the percentage of your capital you should
put at risk.
Most investors understand this criterion
intuitively, at least once it is pointed out. But they follow it insufficiently
if at all. The investment on which they expect a 30% return gets little more
funding than the one where they expect to earn 10%. Investment advantage should
be as important as diversification concerns in determining how one distributes
one’s portfolio.
IV. INVESTING WITH SOMEONE ON THE OTHER SIDE
One of the more puzzling aspects of the
financial world is the volume of transactions in international currency
markets. Average daily volume is $1.9 trillion, which is slightly more than all
U.S. imports in a year. There are hedgers in these markets, to be sure, but
their volume is many times dwarfed by transactions that cross with
sophisticated or at least highly paid traders on both sides. Something no less
magical than levitation is enabling all players to make money, or think that
they are making money.
But let us turn to the
micro situation, where you are trading against a single individual in what may
or may not be a UU situation. If we find that
people make severe mistakes in this arena even when there is merely risk
or uncertainty, we should be much more concerned, at least for them, when UU
may abound.
Bazerman-Samuelson example
and lessons. Let us posit that you are 100% sure
that an asset is worth more to you than to the person who holds it, indeed 50%
more. But assume that she knows the true value to her, and that it is uniformly
distributed on [0,100], that is, her value is equally likely to be 0, 1, 2, …
100. In a famous game due to Bazerman and Samuelson (1983), hereafter BS, you
are to make a single bid. She will accept if she gets more than her own value.
What should you bid?
When asked in the
classroom, typical bids will be 50 or 60, and few will bid as low as 20.
Students reason that the item will be worth 50 on average to her, hence 75 to
them. They bid to get a tidy profit. The flaw in the reasoning is that the
seller will only accept if she will make a profit. Let’s make you the bidder.
If you offer 60, she will not sell if her value exceeds 60. This implies that
her average value conditional on selling will be 30, which is the value of the
average number from 0 to 60. Your expected value will be 1.5 times this amount,
or 45. You will lose 15 on average, namely 60-45, when your bid is accepted. It
is easy to show that any positive bid loses money in expectation. The moral of
this story is that people, even people in decision analysis and finance
classrooms, where these experiments have been run many times, are very poor at
taking account of the decisions of people on the other side of the table.
There is also a strong
tendency to draw the wrong inference from this example, once its details are
explained. Many people conclude that you should never deal with someone else
who knows the true value, when you know only the distribution. In fact, BS
offer an extreme example, almost the equivalent of an optical illusion. You
might conclude that when your information is very diffuse and the other side
knows for sure, you should not trade even if you have a strong absolute
advantage.
That conclusion is wrong.
For example, if the seller’s true value
is uniform on [1,2] and you offer 2, you will buy the object for sure,
and its expected value will be 1.5 times 1.5 = 2.25. The difference between this
example and the one with the prior on [0,1] is that here the effective
information discrepancy is much smaller. To see this, think of a uniform
distribution from [100,101]; there is virtually no discrepancy. (In fact,
bidding 2 is the optimal bid for the [1,2] example, but that the extreme bid is
optimal also should not be generalized.)
Drawing inferences from
others. The general lesson is that people are
naturally very poor at drawing inferences from the fact that there is a willing
seller on the other side of the market. Our instincts and early training lead
us not to trust the other guy, because his interests so frequently diverge from
ours. If someone is trying to convince you that his second hand car is
wondrous, skepticism and valuing your own information highly helps. However, in
their study of the heuristics that individuals employ to help them make
decisions, Tversky and Kahneman (1974) discovered that individuals tend to
extrapolate heuristics from situations where they make sense to those where they
do not.
For example, we tend to
distrust the other guy’s information even when he is on our side. This tendency
has serious drawbacks if you consider sidecar investing – free riding on the
superior capability of others – as we do below. Consider two symmetrically-situated
partners with identical interests who start with an identical prior
distribution about some value which is described by a two- parameter
distribution. They each get some information on the value. They also have identical prior distributions
on the information that each will receive. Thus, after his draw, each has a
posterior mean and variance. Their goal is to take a decision whose payoff will
depend on the true value. The individuals begin by submitting their best
estimate, namely their means. After observing each other’s means, they then
simultaneously submit their new best estimate. Obviously, if one had a tight (loose) posterior his
estimate would shift more (less) toward that of his partner. In theory, two
things should happen:
(a) The two partners should jump over each other
between the first
and second submission half of the time.
(b) The two partners should give precisely
the same estimate for the third submission.
In practice, unless the
players are students of Robert Aumann27 – his article “Agreeing to
Disagree” (1976) inspired this example – rarely will they jump over each other.
Moreover, on the third submission, they will not come close to convergence.
The moral of this story is
that we are deeply inclined to trust our own information more than that of a
counterpart, and are not well trained to know when this makes good sense, and
when it inclines us to be a sucker. One should also be on the lookout for
information disparities. Rarely are they revealed through carnival-barker
behavior. For example, when a seller merely offers you an object at a price, or
gets to accept or reject when you make a bid (as with BS), he will utilize
information that you do not possess. You had better be alert and give full
weight to its likely value, e.g., how much the object is worth on average were
he to accept your bid.
In the financial world one
is always playing in situations where the other fellow may have more
information and you must be on your guard. But unless you have a strictly dominant
action – i.e., it is superior no matter what the other guy’s information -- a
maximin strategy will almost always push you never to invest. After all, his
information could be just such to lead you to lose large amounts of money.
Two rays of light creep
into this gloomy situation: First, only rarely will his information put you at
severe disadvantage. Second, it is extremely unlikely that your counterpart is
playing anything close to an optimal strategy. After all, if it is so hard for
you to analyze, it can hardly be easy for him.28
Absolute advantage and
information asymmetry. It is helpful to break down
these situations into two components. A potential buyer’s absolute advantage
benefits both players. It represents the usual gains from trade. In many financial situations, as we observed above, a
buyer’s absolute advantage stems from her complementary skills. An empty lot in
A’s hands may be worth much less than it would be in B’s. Both gain if A trades
to B, due to absolute advantage. But such an argument would not apply if A was
speculating that the British pound would fall against the dollar when B was
speculating that it would rise. There is no absolute advantage in such a
situation, only information asymmetries.
If both parties recognize
a pure asymmetric information situation, only the better informed player should
participate. The appropriate drawing of inferences of “what-
you-know-since-you-are-willing-to-trade” should lead to the well known no-trade
equilibrium. Understanding this often leads even ordinary citizens to a shrewd strategem:
Maxim C: When information asymmetries may
lead your counterpart to be concerned about trading with you, identify for her
important areas where you have an
absolute advantage from trading. You can also identify her absolute advantages,
but she is more likely to know those already.
When you are the buyer,
beware; seller-identified absolute advantages can be chimerical. For example,
the seller in the bazaar is good at explaining why your special characteristics
deserve a money-losing price – say it is the end of the day and he needs money
to take home to his wife. The house seller who does not like the traffic noise in
the morning may palter that he is moving closer to his job, suggesting absolute
advantage since that is not important to you. Stores in tourist locales are
always having “Going Out of Business Sales.” Most swindles operate because the
swindled one thinks he is in the process of getting a steal deal from someone else.
If a game theorist had
written a musical comedy, it would have been Guys and Dolls, filled as it is with the ploys and plots of
small-time gamblers. The overseer of the roving craps game is Nathan Detroit.
He is seeking action, and asks Sky Masterson – whose good looks and gambling
success befit his name – to bet on yesterday’s cake sales at Lindy’s, a famed
local deli. Sky declines and recounts a story to Nathan:
On the day when I left home to make my way in
the world, my daddy took me to one side. “Son,” my daddy says to me, “I am
sorry I am not able to bankroll you to a large start, but not having the
necessary lettuce to get you rolling, instead I'm going to stake you to some
very valuable advice. One of these days in your travels, a guy is going to show
you a brand-new deck of cards on which the seal is not yet broken. Then
this guy is going to offer to bet you that he can make the jack of spades jump
out of this brand-new deck of cards and squirt cider in your ear. But, son, do
not accept this bet, because as sure as you stand there, you're going to wind
up with an ear full of cider.”
In the financial world at least, a key
consideration in dealing with UU situations is assessing what others are likely
to know or not know. You are unlikely to have mystical powers to foresee the
unforeseeable, but you may be able to estimate your understanding relative to
that of others. Sky’s dad drew an inference from someone else’s willingness to
bet. Presumably Ricardo was not a military expert, but just understood that
bidders would be few and that the market would overdiscount the UU risk.
Competitive knowledge,
uncertainty, and ignorance. Let us assume that you
are neither the unusually skilled Buffett nor the unusually clear-thinking
Ricardo. You are just an ordinary investor who gets opportunities and
information from time to time. Your first task is to decide into which box an
investment decision would fall. We start with unknown probabilities.
Investing with Uncertainty
and Potential Asymmetric Information
|
|
Easy
for Others to Estimate
|
Hard
for Others to Estimate
|
|
Easy for You to Estimate
|
A.
Tough
markets
|
B.
They’re
the Sucker
|
|
Hard for You to Estimate
|
C. Sky Masterson’s Dad, You’re the Sucker
|
D. Buffett’s Reinsurance Sale Calif.
Earthquake Auth.
|
The first row is welcome
and relatively easy, for two reasons:
(1) You probably have reasonable judgment
of your knowledge relative to others, as would a major real estate developer
considering deals in his home market. Thus you
would have a good assessment of how likely you are to be in Box B or Box A.
(2) If you are in Box B, you have the
edge. Box A is the home of the typical thick financial market, where we tend to
think prices are fair on average.
The second row is more
interesting, and brings us to the subject matter of this paper. In Part V
below, we will see Buffett sell a big hunk of reinsurance because he knew he
was in box D. His premium was extremely favorable, and he knew that the likelihood of extreme
odds-shifting information being possessed by the other side was thin. Box C
consists of situations where you know little, and others may know a fair
amount. The key to successfully dealing with situations where you find
probabilities hard to estimate is to be able to assess whether others might be
finding it easy.
Be sensitive to telling
signs that the other side knows more, such as a smart person offering too
favorable odds. Indeed, if another sophisticated party is willing to bet, and he
can’t know that you find probabilities hard to estimate, you should be
suspicious. For he should have reasonable private knowledge so as to protect
himself. The regress in such reasoning is infinite.
Maxim D: In a situation where probabilities
may be hard for either side to assess, it may be sufficient to assess your
knowledge relative to the party on the other side (perhaps the market).
Let us now turn to the more extreme
case, situations where even
the states of the world are
unknown, as they
would be for an angel
investment in a completely
new technology, or for insuring
infrastructure against terrorism over a long period.
Investing with Ignorance and
Potential Asymmetric Information
|
|
Known
to Others
|
Unknown
to Others
|
|
Unknown
to You
|
E. Dangerous Waters Monday Morning Quarterback
Risk
|
F.
Low
Competition Monday Morning Quarterback Risk
|
In some ignorance
situations, you may be confident that others know no better. That would place
you in Box F, a box where most investors get deterred, and where the Buffetts
of this world, and the Rothschilds of yesteryear have made lots of money.
Investors are deterred because they employ a heuristic to stay away from UU situations, because they might
be in E, even though a careful assessment would tell them that outcome was
highly unlikely. In addition, both boxes carry the Monday Morning Quarterback
(MMQ) risk; one might be blamed for a poor outcome if one invests in ignorance,
when it was a good decision that got a bad outcome; might not have allowed for
the fact that others might have had better knowledge when in fact they didn’t;
or might not have allowed for the fact that others might have had better
knowledge, when in fact they did, but that negative was outweighed by the
positive of your absolute advantage. The criticisms are unmerited. But since significant losses were
incurred, and knowledge was scant, the investment looks
foolish in retrospect to all but the most sophisticated. An investor who could
suffer significantly from any of these critiques might well be deterred from
investing.
Let us revisit the Gazprom
lesson within this thought in mind. Suppose you are a Russia expert. It is
still almost inevitable that real Russians know much more than you. What then
should you do? The prudent course, it would seem, would be first to determine
your MMQ risk. It may actually be reduced due to your largely irrelevant
expertise. But if MMQ is considerable, steer clear. If not, and Russian
insiders are really investing, capitalize on Box E, and make that sidecar investment.
You have the additional advantage that few Westerners will be doing the same,
and they are your prime competition for ADRs.29
Speculation 3. UU situations offer great
investment potential given the combination of information asymmetries and lack
of competition.
Boxes E and F are also the
situations where other players will be attempting to take advantage of us and,
if it is our inclination, we might take advantage of them. This is the area
where big money changes hands.
A key problem is to determine
when you might be played for a sucker. Sometimes this is easy. Anyone who has
small oil interests will have received many letters offering to buy, no doubt
coming from people offering far less than fair value. They are monopsonists
after all, and appropriately make offers well below the market. They may not
even have any inside knowledge. But they are surely taking advantage of the
impulsive or impatient among us, or those who do not understand the concepts in
this paper.
Being a possible sucker
may be an advantage if you can gauge the probability. People are strongly
averse to being betrayed. They demand much stronger odds when a betraying human
rather than an indifferent nature would be the cause of a loss (Bohnet and
Zeckhauser, 2004). Given that, where betrayal is a risk, potential payoffs will be too high
relative to what rational decision analysis would prescribe.
Investing in UU with
potentially informed players on the other side. Though
you may confront a UU situation, the party or parties on the other side may be
well informed. Usually you will not know whether they are. Gamblers opine that
if you do not know who the sucker is in
a game that you are the sucker. That does not automatically apply with UU
investments. First, the other side may also be uninformed. For example, if you
buy a partially completed shopping center, it may be that the developer really did run out of money (the proffered explanation for its status) as opposed to
his discovery of deep tenant reluctance. Second, you may have a complementary
skill, e.g., strong relations with WalMart, that may give you a significant
absolute advantage multiple.
The advantage multiple
versus selection formula. Let us simplify and leave
risk aversion and money management matters aside. Further posit, following BS,
that you are able to make a credible take-it-or-leave-it offer of 1. The value
of the asset to him is v, an unknown quantity. The value to you is av, where a
is your absolute advantage. Your subjective prior probability distribution on v
is f (v). The mean value of your prior
is m < 1.30 In a stripped-down model, three parameters describe
this situation: your advantage multiple, a; the probability that the other side
is informed, p; and the selection factor against you, s, if the other side is
informed.31 Thus s is the fraction of expected value that will
apply, on average, if the other side is informed, and therefore only sells when
the asset has low value to her. Of course, given the UU situation, you do not
know s, but you should rely on your mean value of your subjective distribution
for that parameter.
If you knew p = 0, that
the other side knew no more than you, you would simply make the offer if am
> 1. If you knew there were selection, i.e., p = 1, you would invest if your
multiple more than compensated for selection, namely if ams
> 1. The general formula is that your
return will be:
am[ps + (1-p)1] . (1)
Maxim E: A significant absolute advantage
offers some protection against potential selection. You should invest in a UU
world if your advantage multiple is
great, unless the probability is high the other side is informed and if, in
addition, the expected selection factor is severe.
Following Maxim E, you should make your offer
when the expression in (1) exceeds 1.
In practice, you will have
a choice of offer, t. Thus, s will vary with t, i.e., s(t).32 The
payoff for any t will be
If at the optimal offer t*, this quantity is
positive, you should offer t*.
Playing the advantage
multiple versus selection game. Our formulation
posited a take-it-or-leave-it offer with no communication. In fact, most
important financial exchanges have rounds of subtle back-and-forth discussion.
This is not simply cheap talk. Sometimes real information is provided, e.g.,
accounting statements, geological reports, antique authentications. And offers
by each side reveal information as well. Players on both sides know that
information asymmetry is an enemy to both, as in any agency problem.
It is well known that if
revealed information can be verified, and if the buyer knows on what dimensions
information will be helpful, then by an unraveling argument all information
gets revealed.33 Consider a one-dimension case where a value can be
between 1 and 100. A seller with a 100 would surely reveal, implying the best
unrevealed information would be 99. But then the 99 would reveal, and so on
down through 2.
When the buyer is in a UU
situation, unraveling does not occur, since he does not know the relevant
dimensions. The seller will keep private unfavorable information on dimensions
unknown to the buyer. She will engage in
signposting: announcing favorable information, suppressing unfavorable.34
The advantage multiple
versus selection game will usually proceed with the seller explaining why she
does not have private information, or revealing private information indicating
that m and a are large. Still, many favorable deals will not get done, because
the less informed party can not assess what it does not know. Both sides lose
ex ante when there will be asymmetry on common value information, or when, as
in virtually all UU situations, asymmetry is suspected.
Auctions as UU games. Auctions have exploded as mechanisms to sell everything from the
communications spectrum to corporate securities. Economic analyses of auctions
– how to conduct them and how to bid – have exploded alongside. The usual
prescription is that the seller should reveal his information about elements
that will affect all buyers’ valuations, e.g., geologic information on an oil
lease or evidence of an antique’s pedigree, to remove buyers’ concerns about
the Winner’s Curse. The Winner’s Curse applies when an object, such as an oil
lease, is worth roughly the same to all. The high bidder should be aware that
every other bidder thought it was worth less than he did. Hence, his estimate
is too high, and he is cursed for winning.
Real world auctions are
often much more complex. Even the rules of the game may not be known. Consider
the common contemporary auction phenomenon, witnessed often with house sales in
hot markets, and at times with the sale of corporations.35 The
winner, who expected the final outcome to have been determined after one round
of bidding, may be told there will be a best and final offer round, or that now
she can negotiate a deal for the item.
Usually the owner of the
object establishes the rules of the game. In theory, potential buyers would
insist that they know the rules. In practice, they often have not. When
Recovery Engineering, makers of PUR water purifiers, was sold in 1999, a “no
one knows the rules” process ensued, with Morgan Stanley representing the
seller. A preliminary auction was held on an August Monday. Procter and Gamble
(P&G) and Gillette bid, and a third company expressed interest but said it
had difficulties putting its bid together. Gillette’s bid was $27 per share;
P&G’s was $22. P&G was told by the investment banker that it would have
to improve its bid substantially. Presumably, Gillette was told little, but drew
appropriate inferences, namely that it was by far high. The final auction was
scheduled for that Friday at noon. Merrill Lynch, Gillette’s investment banker,
called early on Friday requesting a number of additional pieces of due
diligence information, and requesting a delay till Monday. Part of the
information was released – Gillette had had months to request it – and the
auction was delayed till 5 p.m. Friday. P&G bid $34. At 5 p.m., Merrill
Lynch called, desperate, saying it could not get in touch with Gillette. Brief
extensions were granted, but contact could not be established. P&G was told
that it was the high bidder. Over the weekend a final deal was negotiated at a
slightly higher price; the $300 million deal concluded. But would there have
been a third round of auction if Gillette had bid $33.50 that Friday? No one knows.
The Recovery board puzzled
over the unknowable question: What happened to Gillette? One possibility was
that Gillette inferred from the fact that it was not told its Monday bid was
low that it was in fact way above other bidders. It was simply waiting for a deal to be announced, and then would propose a price perhaps $2 higher, rather
than bid and end up $5 higher.36 Gillette never came back. A while
later, Recovery learned that Gillette was having – to that time unreported –
financial difficulties. Presumably, at the moment of truth Gillette concluded
that it was not the time to purchase a new business. In short, this was a game
of unknowable rules, and unknowable strategies.37 Not unusual.
At the close of 2005,
Citigroup made the winning bid of about $3 billion for 85% of the Guangdong
Development Bank, a financially troubled state-owned Chinese bank. As the New York Times reported the deal, it
“won the right to negotiate with the bank to buy the stake.” If successful
there, its “control might allow Citigroup to install some new management and
have some control over the bank’s future…one of the most destitute of China’s
big banks…overrun by bad loans.”38 Citigroup is investing in a UU
situation, and knows that both the rules
of the game and what it will win are somewhat undefined. But it is
probably confident that other bidders were no better informed, and that both
the bank and the Chinese government (which must approve the deal) may also not
know the value of the bank, and were eager to secure foreign control. Great
value may come from buying a pig in a poke, if others also can not open the bag.
Ideal investments with high
and low payoffs. In many UU situations, even the
events associated with future payoff levels – for example, whether a technology
supplier produces a breakthrough or a new product emerges – are hard to
foresee. The common solution in investment deals is to provide for
distributions of the pie that depend not on what actually happens, but solely
on money received. This would seem to simplify matters, but even in such
situations sophisticated investors frequently get confused.
With venture capital in
high tech, for example, it is not uncommon for those providing the capital to
have a contractual claim to all the assets should the venture go belly up.
Similarly, “cram down” financings, which frequently follow when startups
underperform, often gives VCs a big boost in ownership share. In theory, such
practices could provide strong incentives to the firm’s managers. In reality,
the managers’ incentives are already enormous. Typical VC arrangements given
bad outcomes cause serious ill will, and distort incentives – for example, they
reward gambling behavior by managers after a bleak streak. Worse still for the
VCs, they are increasing their share of the company substantially when the company is not worth much.
They might do far better if arrangements specified that they sacrifice
ownership share if matters turn out poorly, but gain share if the firm does
particularly well.
Maxim F: In UU situations, even sophisticated
investors tend to underweight how strongly the value of assets varies. The goal
should be to get good payoffs when the value of assets is high.
No doubt Ricardo also took Maxim F into
account when he purchased the “Waterloo bonds.” He knew that English money
would be far more valuable if Wellington was victorious and his bonds soared in
value, than if he lost and the bonds plummeted.
A UU investment problem. Now for a harder decision. Look at the letter in Exhibit A, which
offers you the chance to make a modest investment in an oil well. You have
never heard of Davis Oil and the letter came out of the blue, but you inquire
and find out that it is the company previously owned by the famous, recently
deceased oilman Marvin Davis. Your interest is offered because the Davis
Company bought the managing partner’s interest in the prospect from a good
friend and oil man who invited you into his prospect.39 Davis is
legally required to make this offer to you. Decide whether to invest or merely
wait for your costless override before you read on.
Here is what your author did. He started by
assessing the situation. Davis could not exclude him, and clearly did not need
his modest investment. The letter provided virtually no information, and was
not even put on letterhead, presumably the favored Davis approach if it were
trying to discourage investment. Davis had obviously spent a fair amount of
effort determining whether to drill the well, and decided to go ahead. It must
think its prospects were good, and you would be investing as a near partner.
Bearing this in mind, he
called Bill Jaqua – a contact Davis identified in the letter – and asked about
the well. He was informed it was a pure wildcat, and that it was impossible to
guess the probability of success. Some geologic
technical discussion followed, which he tried to pretend he understood.
He then asked what percent of Davis wildcat wells had been successful in recent
years, and got a number of 20-25%. He then asked what the payoff was on average
if the wells were successful. The answer
was 10 to 1. Beyond that, if this well was successful, there would be a number
of other wells drilled in the field. Only participation now would give one the
right to be a future partner, when presumably the odds would be much more
favorable. This appeared to be a reasonably favorable investment, with a
healthy upside option of future wells attached. The clinching argument was that
Jaqua courteously explained that Davis would be happy to take his interest and
give him the free override, thus reinforcing the message of the uninformative
letter not placed on letterhead. (It turned out that the override would have
only been 1% of revenue -- an amount not mentioned in the letter – as opposed
to 76% if he invested.)40 In short, the structure of the situation,
and the nature of Davis’s play made a sidecar investment imperative. The well
has not yet been started.
Davis was in a tough
situation. It had to invite in undesired partners on favorable terms when it
had done all the work. It reversed the usual ploy where someone with a
significant informational advantage tries to play innocent or worse, invoke
some absolute advantage story. Davis tried to play up the UU aspect of the
situation to discourage participation.
Review of the bidding. You have been asked to address some decision problems. Go back now and grade yourself first on the
overconfidence questionnaire. The answers are in the footnote.41
You were asked about three
investments: Tengion, Gazprom and Davis Oil. Gazprom has done nicely over a
six-month period. Neither of the other outcomes has been determined. Go back
and reconsider your choices, and decide whether you employed the appropriate
principles when making them, and then assess the more general implications for
investment in UU situations. Though this essay pointed out pitfalls with UU investing,
it was generally upbeat about the potential profits that reside in UU arenas.
Hopefully you have been influenced, at least a
bit.
V. A BUFFETT TALE
The following story encapsulates the fear of
UU situations, even by sophisticated investors, and the potential for shrewd
investors to take great advantage of such situations. In 1996, I was attending
an NBER conference on insurance. One participant was the prime consultant to
the California Earthquake Authority. He had been trying to buy a $1 billion
slice of reinsurance – to take effect after $5 billion in aggregate insured
losses -- from the New York financial community. The Authority was offering
five times estimated actuarial value, but had no takers. It seemed exceedingly
unlikely that the parties requesting coverage had inside information that a
disastrous earthquake was likely. Hence, there was a big advantage, in effect a
= 5, and p was close to 0. Maxim E – weigh absolute advantage against
informational disadvantage – surely applied.
My dinner table syndicate
swung into action, but ended up $999.9 million short. A couple days later, we
learned that Buffett had flown to California to take the entire slice. Here is
his explanation.
…we wrote a policy for the California
Earthquake Authority that goes into effect on April 1, 1997, and that exposes
us to a loss more than twice that
possible under the Florida contract. Again we retained all the risk for our own
account. Large as these coverages are, Berkshire's after- tax
"worst-case" loss from a true mega-catastrophe is probably no more
than $600 million, which is less than 3% of our book value and 1.5% of our
market value. To gain some perspective on this exposure, look at the table on
page 2 and note the much greater volatility that security markets have
delivered us. [Chairman’s letter to the Shareholders of Berkshire Hathaway, 1996, http://www.ifa.com/Library/Buffet.html]42
Reinsurance for earthquakes is certainly a
venture into the unknown, but had many attractive features beyond its dramatic
overpricing. Unlike most insurance, it was exceedingly unlikely that the
parties taking insurance had inside knowledge on their risk. Thus,
Buffett – despite attention to money management -- was willing to take 100% of
a risk of which Wall Street firms houses rejected taking even part. Those fancy
financial entities were not well equipped to take a risk on something that was
hard for them to estimate. Perhaps they did not recognize that others had no
inside information, that everyone was operating with the same probability. And
perhaps they were just concerned about Monday Morning Quarterbacking.
It is also instructive to
consider Buffett’s approach to assessing the probabilities in this UU situation,
as revealed in the same annual report:
So what are the true odds of our having to
make a payout during the policy's term? We don't know - nor do we think
computer models will help us, since we believe the precision they project is a
chimera. In fact, such models can lull decision-makers into a false sense of
security and thereby increase their chances of making a really huge mistake.
We've already seen such debacles in both insurance and investments. Witness
"portfolio insurance," whose destructive effects in the 1987 market
crash led one wag to observe that it was the computers that should have been
jumping out of windows.
Buffett was basically saying to Wall Street
firms: “Even if you hire 100 brilliant Ph.D.s to run your models, no sensible
estimate will emerge.” These are precisely the types of UU situations where the
competition will be thin, the odds likely favorable, and the Buffetts of this
world can thrive.
As Buffett has shown on
repeated occasions, a multi-billionaire will rush in where mathematical wizards
fear to tread. Indeed, that explains much of his success. In 2006 hurricane
insurance met two Buffett desiderata, high prices and reluctant competitors. So
he plunged into the market:
Buffett’s prices are as much as 20 times
higher than the rates prevalent a year ago, said Kevin Madden, an insurance
broker at Aon Corp. in New York. On some policies, premiums equal half of its
maximum potential payout, he said. [In a May 7, 2006, interview Buffett said:]
“We will do more than anybody else if the price is right… We are certainly
willing to lose $6 billion on a single event. I hope we don’t.'’ http://seekingalpha.com/article/11697
At least two
important lessons emerge from thinking about the “advantage-versus-
Maxim G: Discounting for ambiguity is a
natural tendency that should be overcome, just as should be overeating.
Maxim H: Do not engage in the heuristic
reasoning that just because you do not know the risk, others do. Think
carefully, and assess whether they are likely to know more than you. When the
odds are extremely favorable, sometimes it pays to gamble on the unknown, even
though there is some chance that people on the other side may know more than you.
Buffett took another bold
financial move in 2006, in a quite different field, namely philanthropy. He
announced that he would give away 85% of his fortune or $37.4 billion, with $31
billion going to the Bill and Melinda Gates Foundation. Putting money with the
Gates Foundation represents sidecar philanthropy. The Foundation is an
extremely effective organization that focuses on health care and learning. It
is soon to be led by Bill Gates, a fellow with creativity, vision and
hardheadedness as strong complementary skills, skills which are as valuable in
philanthropy as they are in business.
VI. CONCLUSION
This essay offers more speculations than
conclusions, and provides anecdotal accounts rather than definitive data. Its
theory is often tentative and implicit. But the question it seeks to answer is
clear: How can one invest rationally in UU situations? The question sounds
almost like an oxymoron. Yet clear thinking about UU situations, which includes
prior diagnosis of their elements, and relevant practice with simulated
situations, may vastly improve investment decisions where UU events are
involved. If they do improve, such clear thinking will yield substantial
benefits. For financial decisions at least, the benefits may be far greater
than are available in run-of-the-mill contexts, since competition may be
limited and prices well out of line.
How important are UU
events in the great scheme of financial affairs? That itself is a UU question.
But if we include only those that primarily affect individuals, the magnitude
is far greater than what our news accounts would suggest. Learning to invest
more wisely in a UU world may be the most promising way to significantly
bolster your prosperity.
APPENDIX A
Assessing
Quantities*
1. Democratic votes
in Montana, 2004 Presidential election
2.
Length of Congo River (in miles)
3.
Number of subscribers to Field and Stream
4.
Area of Finland (in square miles)
5.
Birth rate in France per 1,000 population
6.
Population of Cambodia
7.
Revenues of Wal-Mart Stores (largest in U.S.), 2003
8. Annual Percent Yields on
30-Year Treasury Bonds in 1981 (This year had the highest rate over the 1980-1998 period.)
9.
Number of physicians in the United States, 2002
10. Number of
electoral votes going to Republican presidential candidate in 2008 (out of 538)
11. Value of Dow
Jones Average on December 31, 2006 (on 6/30/06 closed at 11,150)
12. Value of the
NASDAQ on December 31, 2006 (on 6/30/06 closed at 2,172)
|
|
1st
%ile
|
25th
%ile
|
50th
%ile
|
75th
%ile
|
99th
%ile
|
|
Democratic votes MT
2004 Pres. election
|
|
|
|
|
|
|
Congo River
(length in miles)
|
|
|
|
|
|
|
Field &
Stream (number
of subscribers)
|
|
|
|
|
|
|
Finland
(area in square miles)
|
|
|
|
|
|
|
Birth Rate of France
(per thousand)
|
|
|
|
|
|
|
Population of Cambodia
|
|
|
|
|
|
|
Revenues of Wal-Mart
Stores, 2003
|
|
|
|
|
|
|
% Yields on
30-Year Bonds, 1981
|
|
|
|
|
|
|
Number of Physicians in
U.S., 2002
|
|
|
|
|
|
|
# electoral college votes,
Republican presidential candidate in 2008
|
|
|
|
|
|
|
Dow Jones Average 12/31/06
(on 6/30/06 closed at 11,150)
|
|
|
|
|
|
|
Value of NASDAQ 12/31/06
(on 6/30/06 closed at 2,172)
|
|
|
|
|
|
Remarks:
∗I thank Miriam Avins, Paul Samuelson and
Nils Wernerfelt for helpful comments.
1 The financing of 36 million pounds was floated on the London Stock
Exchange. Ricardo took a substantial share. His frequent correspondent Thomas
Malthus took 5,000 pounds on Ricardo’s recommendation, but sold out shortly
before news of the Waterloo outcome was received. The evidence is clear that
Ricardo, in his words, understood the “dismal forebodings” of the situation,
including “its consequences, on our [England’s] finances.” See Sraffa (1952,
Vol VI, pp. 202, 229 and surrounding material.
2 Ralph Gomory’s (1995) literary essay on the Unknown and Unknowable
provided inspiration. Miriam Avins provided helpful comments.
3 The classic
description of uncertainty, a situation where probabilities could not be known,
is due to Frank Knight (1921).
4 This is sometimes expressed that things move geometrically rather than
arithmetically, or that the logarithm of price has a traditional symmetric
distribution. The most studied special case is the lognormal distribution. See
“Life is log-normal” by E. Limpert and W. Stahel, http://www.inf.ethz.ch/personal/gut/lognormal/brochure.html, for
an argument on the widespread applicability of this distribution.
5 Ricardo’s major competitors
were the Baring Brothers and the Rothschilds. Do not feel sorry for the
Rothschilds. In the 14 years from 1814 to 1828 they multiplied their money
8-fold, often betting on UU situations, while the Baring Brothers lost capital.
http://www.businessweek.com/1998/49/b3607071.htm.
Analysis based on Niall Ferguson’s House
of Rothschild.
6 Hart and Tauman (2004) show that market crashes are possible purely
due to information processing among market participants, with no new
information. They observe that the
1987 crash – 20% in a day – happened
despite no new important information becoming available, nor negative economic
performance after the crash. Market plunges due to ordinary information
processing defies any conventional explanation, and is surely a UU event.
7 Nassim Taleb and Benoit Mandelbrot posit that many financial phenomena
are distributed according to a power law, implying that the relative likelihood
of movements of different sizes depends only on their ratio. Thus, a 20% market
drop relative to a 10% drop is the same as a 10% drop relative to a 5% drop. http://www.fooledbyrandomness.com/fortune.pdf. Power distributions have fat tails. In their
empirical studies, economists frequently assume that deviations from predicted values have normal
distributions. That makes computations tractable, but evidence suggests that
tails are often much thicker than with the normal. Zeckhauser and Thompson (1970).
8 Complementary skills can also help the less affluent invest. Miriam Avins, a good friend, moved into an edgy neighborhood in Baltimore
because the abandoned house next door looked like a potential community garden,
she knew she had the skills to move the project forward, and she valued the
learning experience the house would bring to her family. Her house value
doubled in 3 years, and her family learned as
well.
9 Dinner speech to annual executive program on Investment Decisions and
Behavioral Finance, John F. Kennedy School of Government, Harvard University,
October 14, 2004.
10 This investment was proposed when this paper was presented at a
conference sponsored by the Wharton School on January 6, 2006. The price was
then 33.60. At press time nine months later it was $47.
12 Approximate average from Investment Decisions and Behavioral Finance,
executive program, annually fall 2001-2006, and API-302, Analytic Frameworks
for Policy course. The former is chaired, the latter taught by Richard
Zeckhauser, Kennedy School, Harvard University.
13 See Gilbert (2006) for insightful discussions of the problems of
rationalization and corrigibility.
14 See Viscusi and
Zeckhauser (2005).
15Kahneman and
Tversky (1979).
16 This illustration employs events that may have happened in the past,
but subjects would not know. The purpose is to make payoffs immediate, since
future payoffs suffer from a different
form of bias.
17 The experiment is at a disadvantage in getting this result, since
peoples’ assessments of the contingencies’ probabilities would vary widely.
Some would pick D because they attached an unusually high probability to it. In
theory, one could ask people their probability estimate after they made their
choice, and then look only at the answers of those for whom the probability was
in a narrow range. However, individuals would no doubt adjust their
retrospective probability estimates to help rationalize their choice.
18 This experiment and the choice between lotteries C and D above only
approximate those with numerical probabilities, since they are calibrated for
median responses and individuals’ estimates will differ.
19 In fact, Ellsberg’s
experiment involved drawing a marble of a particular color from an urn.
Subjects preferred a situation where the percentage of winning marbles was
known, even if they could bet on either side when it was unknown.
20 Fox and Tversky (1995, p. 585) found that ambiguity aversion was
“produced by a comparison with less ambiguous events or with more knowledgeable
people….[it] seems to disappear in a noncomparative context.” Ambiguity
aversion is still relevant for investments, if alternative investments are
available and contemplated.
21 Paul Samuelson, who attends closely to most aspects of the finance
field, attests to this challenge. He observed that the Renaissance Group, run
by former Stony Brook math professor Jim Simons, is “perhaps the only long-time
phenomenal performer [in traditional financial markets] on a risk-corrected
basis.” Private communication, June 15, 2006.
22I saw such path blazing by my former business partner Victor
Niederhoffer in the 1970s, when he ventured into commodity investing. His
associates hand recorded commodity prices at 15- minute intervals. He lined up
a flotilla of TRS-80 Radio Shack computers to parallel process this
information. His innovative data mining, spurred by accompanying theories of
how markets behave, gave him a giant advantage over major investment houses.
Niederhoffer continues along unusual paths, now making a second fortune after
losing his first in the collapse of the Thai baht in 1997.
23 Samuelson, Paul A. (1979). “Why We Should Not Make Mean Log of Wealth
Big Though Years to Act Are Long,” Journal
of Baking and Finance 3: 305-307.
25 In the language of decision theory, individuals who follow Kelly
rather than maximizing expected utility would be making a sacrifice in the certainty
equivalent value of their terminal wealth, i.e., the wealth that results after
participating in a string of gambles. The Kelly criterion is appropriate for
someone with a logarithmic utility function.
26 For example, in real estate, a limited partnership interest that will
come due in a few years is likely to sell about 30% below discounted expected
future value. The significant discount reflects the complementary skills of
acquirers, who must be able to assess and unlock the value of idiosyncratic partnerships.
Personal communication, Eggert Dagbjartsson, Equity Resource Investments,
December 2005. That firm earns substantial excess returns through its
combination of effective evaluation of UU situations, and the unusual
complementary skill of being able to deal effectively with recalcitrant general
partners. Experience with Dagbjartsson’s firm – with which the author is
associated – helped inspire this paper.
27 Robert Aumann and Thomas Schelling won the 2005 Nobel Memorial Prize
in Economics for their contributions to game theory.
28 Given the potential for imperfect play, it is sometimes dangerous to
draw inferences from the play of others, particularly when their preferences
are hard to read. The Iraqi weapons of mass destruction provide a salient
example. Many people were confident that such weapons were present not because
of intelligence, but because they believed Saddam Hussein could have saved
himself and his regime simply by letting in inspectors, who in the instance
would find nothing.
29 In January 2006,
Gazprom traded in the west as an ADR, but soon became an over-the-counter
stock.
30 It is important
that m < 1. Otherwise the seller would refuse your offer if he were
uninformed.
31 In health care,
this process is called adverse selection, with sicker people tending to enroll
in more generous health plans.
32 Let v be
the conditional mean of x < v. The value of s will be constant if v/v
= positive k for all v. This will be the case if
f(v) is homogeneous, i.e., f(kv) = knf(v), as with the uniform or
triangular distribution starting at 0.
33 See Grossman (1981) on unraveling. If information is costly to reveal,
then less favorable information is held back and signposting applies
(Zeckhauser and Marks, 1996).
34 To be sure, the shrewd buyer can deduce: “Given the number of unknown
dimensions I suspected, the seller has revealed relatively few.” Hence, I
assume that there are a number of unfavorable dimensions, etc. When seller
revelation is brief, only high m buyers will make exchanges. The doubly shrewd
buyer may be informed or get informed on some dimension without the seller
knowing which. He can then say: “I have unfavorable information on a dimension.
Unless you reveal on all dimensions, this information will stay private, and I
will know that you are suppressing
information.” The triply shrewd buyer, knowing nothing, will make the same statement. The shrewd seller
has countermeasures, such as insisting on proof that the buyer is informed,
e.g., by third party attestation, and if evidence is received then revealing
some but not all, hoping to hit the lucky dimension.
35 See Subramanian
and Zeckhauser (2004), who apply the term “negotiauctions” to such processes.
36 Recovery created a countermeasure to raise any post-deal bid by
inserting a breakup fee in its deal with P&G that declined (ultimately to
0) with the price premium paid by a new buyer.
37 Details confirmed by Brian Sullivan, then CEO of Recovery Engineering,
in personal communication, January 2006. Zeckhauser was on the Recovery board
due to a sidecar privilege. He had been Sullivan’s teacher, and had gotten him
the job.
38 New York Times, December 31, 2005, B1 and
B4. Citigroup had several Chinese state-owned companies as partners, but they
probably gave more political cover than knowledge of the value of the bank.
39 That man was Malcolm Brachman, president of Northwest Oil, a bridge
teammate and close friend. Sadly Malcolm had died in the interim. One
consequence was that he could not advise
you.
40 Not mentioned in
the letter was that 24% went off the top to priority claims, and that Davis
charges 75% if you take the free override.
41 1) 173,710 2)
2,716 3) 2,007,901 4) 130,119 5) 13 6) 12,212,000 7) $259B 8) 13.45% 9)
853,000,
10) 173, 11) 12,466, 12) 2,444
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