Did you know the world almost ended in June? For real…
The world as we know it may have come within hours of ending on June 29, 2024 when an asteroid, discovered only about one week before, passed closer to the earth than the moon is. A few hours difference in the timing, and the asteroid would have intersected the earth’s orbit and would have struck the earth.
In astronomical terms, that’s a very close shave. [1]
The asteroid has an estimated diameter of 150 meters (think of a rock about the height of a 50-story building). Scientists estimate such an object hitting the earth would release energy equivalent to over five times the energy of the largest nuclear weapon ever exploded. [2],[3]
Risk vs. Uncertainty
We all tend to use the word risk somewhat loosely. In this post, the first of a series, we will explore risk from a number of different points of view. We will start by differentiating risk from uncertainty. We will then look at number of different ways of conceiving of risk. The overall goal is to develop a comprehensive framework for thinking about risk, and structuring our lives to live with the unescapable facts of uncertainty and risk.
Risk, in its many forms, arises because we humans are uncertain about what will, or will not, happen in the future
Uncertainty is a word for the fact that consistent, accurate and indubitable knowledge about the future is not generally available to us.
We may have faith, or belief, that some particular event will (or will not) occur in the future. For example, I would guess that everyone reading this has no meaningful doubt that the sun will rise tomorrow morning. [4]
You might be thinking that the chance of the sun failing to rise, or of a large asteroid striking the earth, are both so small as to be inconsequential.
If so, you are now thinking in terms of risk.
Risk
Twentieth century economist Frank Knight, in his 1921 book Risk, Uncertainty and Profit, distinguished between uncertainty and risk.
Risk, he said, applies to information we’d like to know which is “susceptible of measurement.” “Uncertainty,” he said, “must be taken in a sense radically distinct from the familiar notion of Risk.” [5]
Knight goes on to describe three categories of risk. He identified:
1. A priori probability. Absolutely homogeneous classification of instances completely identical except for really indeterminate factors. This judgment of probability is on the same logical plane as the propositions of mathematics.
2. Statistical probability. Empirical evaluation of the frequency of association between predicates.….
3. Estimates. The distinction here is that there is no valid basis of any kind for classifying instances. This form of probability is involved in the greatest logical difficulties of all, and no very satisfactory discussion of it can be given, but its distinction from the other types must be emphasized and some of its complicated relations indicated. [6]
Probability
Most meaningful discussions of risk must involve the notion of probability. The mathematical treatment of probability goes back several centuries, and the math of probability is well worked out. (Though probability can still be quite challenging to understand and implement correctly in real world situations.)
The main areas of probability we will address include:
a. the interpretation of probabilities
b. Bayesian inference
Interpretation
What Knight calls a priori probabilities are the easiest to interpret. A priori probabilities are purely logical constructs, so we can, by assumption, know everything we need to know about them. Unfortunately, they occur only in mathematical settings.
For example, it is extremely common in probability courses to talk about flipping coins. Sometimes the additional assumption of a “fair” coin is added.
We all “know” that a coin flip has a 50% probability of coming up heads, and an equal 50% probability of coming up tails.
But how do we know these probabilities?
In terms of a priori probabilities, we know that a “fair” coin has equal probabilities on each flip of heads and of tails, because that is the definition of a fair coin.
Fair dice provide another common example of a priori probabilities. By definition, a fair die has a 1/6th probability of coming up 1, 2, 3, 4, 5 or 6.
Empirical Probabilities
In the real world, few probabilities are known with anything approaching a priori certainty. Devices, such as casino games and lotteries, if they are honestly designed, come pretty close.
Consider a real coin, such as a quarter. A great deal of careful experimental evidence [7] suggests that real coins are very close to ideal “fair” coins.
Thus, it is quite reasonable to view the probability of a flipped coin coming up tails at 50% and heads as 50%.
Similarly, real world casino dice are “as close as possible to random dice”, [8] according to researchers at Western Kentucky University.
A Key Assumption: Independent and Identically Distributed (“IID”)
When we say that the probability of a flipped coin coming up heads is 50%, or that the probability of a six-sided die coming up 3 is 1/6th, we are relying on the assumption known as Independent and Identically Distributed, or IID.
This assumption means that the coin, or die, behaves the same every time, and that it has “no memory.” The “no memory” assumption means that there is no predictive information in knowing the history of the coin flips, or die rolls.
Balls and Urns
The metaphor of an urn containing black and white balls serves as a mental model for understanding random draws from a distribution. For example, imagine an urn contains 100 balls, of which 70 are white and 30 are black. Every time a ball is withdrawn at random (assuming there are always 70 white and 30 black), there is a 70% probability it will be white, and 30% that it will be black.
We say that the system that produces the observations (draws) is stable. The draws are considered independent and identically distributed. If the ratio of white to black balls changed between drawings, the draws would not be identically distributed.
Real World Empirical Probabilities
In the real world, most of the probabilities we’re interested in are not as well-behaved as are coins and casino dice. By well-behaved, we mean that the behavior does not correspond as closely to a theoretical a priori probability as do coins and dice.
Weather Forecasting Relies on Empirical Probability
Weather forecasters (as a profession) have gathered an enormous database of past weather patterns. They have studied those patterns, and discovered many regularities that allow them to make very accurate predictions.
By accurate, we mean that if the forecast is for rain tomorrow, it is highly likely to rain tomorrow. Good weather forecasts rarely say “it will rain tomorrow.” Rather, they say something like “there is a 60% chance of rain tomorrow.” We say the forecasts are well calibrated if, for example, the forecast is a 60% chance of rain tomorrow, and over a large number of observations, it in fact does rain about 60% of the time.
Weather forecasts are able to be accurate and well calibrated because the system – the global climate – that produces weather is a pretty stable system. That is, as far as we can tell, each “draw” – i.e. each day’s weathers – occurs as though drawn independently from an identical distribution. [9]
Investment History
Many researchers have investigated the statistical properties of returns generated by investments. For example, the long run returns from the US stock market have averaged about 10% per year (compound annual return) with realized annual standard deviation of about 17.5% per year.
We will have much more to say in future posts about the statistical properties of investment returns.
The point for here is that the historical standard deviation of stock returns is relevant only if we believe that future returns will be drawn (in a statistical sense) from essentially the same distribution that has produced past returns.
Subject to this significant caveat, it is appropriate to say that in Knightian terms, stock risk fits into his category of “statistical risk.”
Estimates
Knight’s third category of risk is what he calls estimates. These are potential events about which we have no meaningful history, or insufficient history, to form a statistically based inference.
This brings us to the important topic of subjective probability, which we will deal with in a separate post.
[1] https://www.jpl.nasa.gov/news/nasas-planetary-radar-tracks-two-large-asteroid-close-approaches/
[2]This webpage from Tulane university has lots of good information for the curious reader. https://www2.tulane.edu/~sanelson/Natural_Disasters/impacts.htm#:~:text=Velocity%20and%20Energy%20Release%20of,impacting%20body%20and%20its%20velocity
[3] Kinetic energy = ½ mv2 where m = mass, and v = velocity. Assuming velocity of 25,000 meters/second, diameter of 150 meters, and density twice that of water, the energy release would be 1.1×1018 joules, which is about the same energy release as 264 megatons of TNT. That’s over five times as large as the largest hydrogen bomb (the Tsar Bomba), and over fifty times all the explosive ordinance used in WWII.
[4] While the sun failing to rise (or the earth failing to rotate) is not something I or most people ever worry about, philosophers and physicists tell us that this disastrous phenomenon is not impossible. E.g. David Hume, considered by some to be the most important philosopher to write in English, raised the doubt in his Enquiry Concerning Human Understanding (1748). Physicists and astrophysicists tell us that it is theoretically possible that the earth could be impacted by an object (such as an asteroid, or even a black hole) with sufficient energy to destroy the earth, or at least all human life on it. If the earth rotates and there’s no one alive to see it, did the sun rise?
[5] Frank H. Knight, Risk, Uncertainty and Profit, Augustus M. Kelly, reprinted in 1964, p. 19.
[6] Ibid, pp. 224-225.
[7] E.g. https://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses.html
[8] https://digitalcommons.wku.edu/seas_faculty_pubs/7/#:~:text=Casino%20dice%20come%20as%20close,as%20the%20plastic%20die%20body
[9] The fact that weather forecasts are as accurate as ever (if not better) would seem to be evidence that the system that produces the weather is stable. That, in turn, would seem to suggest that if climate change is actually occurring, it is occurring in such a way as to not disturb the predictability of the resulting weather.
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