Risk and uncertainty are often associated together. When one contemplates partaking in a risky endeavor, there is typically uncertainty surrounding the probability of an event occurring. Few people would argue that gambling is a riskless endeavor, nor is participating in financial markets. There exists a probability that one loses the money they elect to play or invest with, thus highlighting their inherent risks. But there does exist a large distinction between casino games and financial markets, and it is an issue surrounding probability assignment.
Take, for example
Noticeably, one player is a 67% favorite to win this hand, while the other is a 33% underdog. If this situation occurred an infinite number of times, the player with pocket jacks would win 67% of the time. While the player with jacks cannot be certain she will win any given runout, she can be certain about the frequency at which she will win over the long run because the outcomes of casino games are known and their probabilities are computable.
Some games do not have a well-defined range of outcomes, highlighting the difference between risk and uncertainty. Take, for example, investing in the stock market. Statistically speaking, the returns of asset prices are a non-stationary process, meaning that the distribution defining their returns changes over time. Imagine playing a game in which a die is rolled, and you receive $1 for a number below a four but have to pay $1 for a number greater than or equal to four. The expected value of this game over the long run is $0, and you would expect to see the numbers one, two, and three appear face-up 50 percent of the time. But suppose the person rolling the die switches the die on you, replacing the standard six-sided die with the one-hundred face polyhedral die pictured below.
Now, your expected value for playing this game is -$97 over the long run, and you would expect to observe the numbers one, two, and three only 3 percent of the time! Unbeknownst to you, the probability distribution surrounding the range of outcomes defining this game changed when your nefarious opponent switched the die, and you can no longer be certain about the probability distribution defining the game moving forward since your nemesis could pull out ever-more elaborate die to play this game with.
This is a simple reduction of how financial markets operate, but it encapsulates the distinction between risk and uncertainty. In financial markets, observing more events provides you with more data to perform statistical analysis on, giving one the ability to assign a probability to an event’s occurrence. That is until the distribution defining those outcomes changes. We do not know the number of faces on the die which governs how financial markets operate, and that uncertainty makes investing an incredibly challenging activity. Simultaneously, markets reward the bearers of this uncertainty by providing them with a return on their investment.
The discussion above leads us to several interesting questions:
(1) Is something risky if we do not know what outcome will occur, but can assign probabilities to the occurrence of every potential outcome?
(2) Is something risky if we are uncertain about its probability distribution or its range of outcomes?
(3) And lastly, if there is uncertainty surrounding the range of outcomes, but the distribution governing the outcomes is a favorable one, does that make the activity risky?
Let us examine question (1) in more depth. If we can assign probabilities to the occurrence of potential outcomes, then we can manage our capital more effectively. The Kelly Criterion is an application of probability theory that maximizes a player’s wealth in games in which they have an edge. Ed Thorpe championed the strategy, using it to play blackjack and later to operate in financial markets. In effect, if a player knows he has an edge in a betting game and is aware
As shown in the graph above, the fraction of capital which maximizes a player’s expected terminal wealth is .20 of her bankroll. Thus, if a player has an edge in a game with a defined probability distribution, the uncertainty defining the optimal strategy disappears; one must simply bet the Kelly fraction to maximize his expected long-term wealth, and one could even recommend betting less than that amount in order to lower the probability of ruin. What about risk pertaining to games in which a player does not have an edge? The answer is clear: simply abstain from playing!
Therefore, the main risk involved with games that have defined probability distributions is one of luck. Players can go broke through chance alone, but the role of luck is mitigated by playing in games in which they have an edge and by limiting the size of their bets. Thus, being knowledgeable of adverse outcomes mitigates the “risk” associated with them since one can take precautions against them.
Which brings us to question (2). This is an idea Nassim Taleb has grappled with throughout his Incerto series, specifically in his second installment The Black Swan. The idea perpetuated throughout the book is akin to a statement made by Donald Rumsfeld:
“Reports that say that something hasn’t happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don’t know we don’t know. And if one looks throughout the history of our country and other free countries, it is the latter category that tend to be the difficult ones.”
Indeed, the area of unknown unknowns is the issue we face when dealing with uncertainty regarding an outcome’s distribution since we cannot assign probabilities to an event’s occurrence with certainty nor can we be sure that we are aware of every potential outcome. To illustrate, consider the turkey problem. In this scenario, a turkey is updating its probability of being fed tomorrow given that it received food today and believes that there are only two events that may occur. Namely, the turkey expects that it will either receive food or it will not receive food. On day one, since it is uncertain that it will receive food the next day, it believes the probability of getting fed is 0.5. However, when it receives food on the first day, it becomes more confident it will receive food tomorrow, increasing its probability estimate. This updating process repeats until the turkey’s last day on earth, a day in which an event that it never foresaw occurs. Not only does our turkey fail to receive food; unfortunately for our champion turkey, it receives an ax to the head, resulting in – well you get it. Paradoxically, our turkey’s confidence in obtaining food tomorrow nears its highest point while our turkey approaches the last days of its life, as the plot below shows.
Thus, the uncertainty surrounding the set of outcomes that can occur is the ultimate form of risk, specifically if the unknown outcomes are detrimental to a risk-taker’s endeavors. Taleb’s remedy for such events is to expose yourself to positive black-swans (taking advantage of the unforeseen rise of the internet) and to protect yourself against negative black-swans (buying hurricane insurance for your coastal home). Although unknown positive outcomes may exist, that does not eliminate the threat posed by unknown negative outcomes. It is imperative to protect yourself from these unknown negative events since they can lead to a blowup, removing you permanently from your endeavor as a result.
What about question (3), a situation in which we are uncertain about the distribution of outcomes, but the distribution turns out to be a favorable one? Entrepreneurship is often deemed to be a risky endeavor, yet I think it is perceived as risky because of the uncertainty surrounding a startup’s distribution. Since there is no historical distribution to base your expectations on, a risk-taker in a new field slowly uncovers the true distribution of outcomes through trial and error. Opportunities with favorable outcomes should be pursued further while those with adverse distributions should be discarded from your pool of ideas. This is the basis on which many successful entrepreneurs operate: they continue to pursue good ideas and cut losses on their bad ones. Similarly, successful traders let their winning trades compound and cut losses on their bad trades.
The process of uncovering an unknown distribution is akin to discovering a fossil. Archeologists chip away at the earth covering the fossil, revealing a beautiful discovery. So, if the probability of an entrepreneur’s success was high – conditional upon her decision to pursue that idea – was it a risky decision? Or did the uncertainty which shrouded the underlying distribution give the endeavor the illusion of risk?
It is difficult to know which ideas will have favorable distributions a priori, especially if you are the first person to pursue them. However, bearing the uncertainty associated with such a decision comes with handsome economic rewards: that of equity, and potentially scale. These are perks that few people in the corporate realm ever obtain. Philosophically, corporate jobs raise your floor at the expense of capping your upside. Corporate employees trade convexity and nonlinearity for security and structure. Since the path to the corporate world has been traveled by many others, there exists a well-defined distribution of outcomes. Additionally, there often exists a systematic procedure for obtaining a job. Conversely, pursuing a new idea is inherently uncertain, since there is no referential mechanism available to you. An entrepreneur in a new frontier is on their own, navigating through the unknown in search of their idea’s probability distribution. They are archeologists sifting through the uncharted sand, searching for an undiscovered fossil. Thus, we have stumbled upon the missing quadrant of Donald Rumsfeld’s two by two matrix. We have defined the unknown knowns, endeavors which are likely to succeed conditional upon discovery of their governing distribution.
A hole regarding my answer to question (3) lies in its apparent disregard of hindsight bias. Hindsight bias illuminates the issue that entrepreneurs could be perceived as successful simply because they got lucky. Indeed, one may be prone to rationalize the success of an endeavor simply because it was successful, regardless of the risk entailed in obtaining that success. To illustrate, examine the following distribution, named after Taleb who popularized the idea.
This distribution is representative of the turkey problem that was defined earlier. Specifically, the average outcome is a good one, but off in the tail of this distribution there exists a risk of systemic failure. If success is found when operating in a distribution such as the one above, there was luck involved in the outcome as the risk-taker avoided the adverse outcome of a blowup. The green Xs denote the risk-taker’s observed outcomes; four of his payouts were positive ones, and one was a negative one which did not result in a blowup. If this game was played an infinite number of times, our risk-taker would eventually walk away with nothing since the left
Now take the following distribution whose mean is the black line, and where events falling to the left of the red line denote bad outcomes (ignore the axes).
This is representative of the positive distribution governing our unknown knowns; on average good outcomes will occur, and there exists a fat tail on the right side of the distribution, opposite to the left tail which exists in the turkey problem. Clearly, entrepreneurs navigating within this distribution will experience positive outcomes in the long run. But unfortunately, we still cannot escape luck in its entirety. Take the two following scenarios where the green Xs denote observed outcomes.
If we assume our entrepreneur is cutting her losses while pursuing her winners, then the entrepreneur in Scenario 1 would likely pursue a different opportunity while the entrepreneur in Scenario 2 would continue to pursue this one. In effect, bad luck would cause entrepreneur 1 to fold a favorable distribution, while entrepreneur 2 continues to reap its rewards. Venture capital funds benefit from this scenario by diversifying their bets among entrepreneurs competing within the same favorable distribution. This enables them to mitigate the variability associated with the sampling process. Some unlucky entrepreneurs they invest in may fold a good hand, but entrepreneurs who receive positive feedback from the outset of their endeavor continue to profit from a favorable distribution of outcomes, enabling venture capital funds to profit accordingly since the expected value of the sum of their bets is positive.
The main issue with our unknown knowns, then, is ensuring one can reap the rewards of distributions with favorable outcomes. Entrepreneurs who get unlucky may fold a good distribution, but that does not mean they will fail to find another. If people are able to think critically about the characteristics governing their opportunity set, while managing their capital effectively, this ensures they can keep skin in the game while tinkering on the edge of the unknown until they
In life, we are too quick to mistake uncertainty for risk without conducting thoughtful analysis regarding a potential action. In this piece, we attempted to differentiate between risk and uncertainty by asking three distinct questions. I have offered my answers and I urge the reader to take the time to answer them as well.
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