Financial market participants struggle with tail risk scenarios. This is an undeniable fact, going back to the days of Black Friday, LTCM, and Lehman. This has recently come to focus last week, after a “multiple sigma” day, as measured by the behavior of various *factors* that explain security movement. These “abnormal” moves have become increasingly commonplace, like 100 year storms that happen to hit our coasts on a bi-annual basis. This chart did its rounds late last week, highlighting the preponderance of single-day CBOE VIX spikes in the last decade:

Why is the recent trend bucking expectations? Is the world changing to an extent that historical risk metrics are invalid? Perhaps; we’re not entirely sure. What we do know is the statistical oversimplifications we engage in to describe market phenomenon are wrong. We have weaponized this oversimplification through the development of tooling and systems of automation that can compound error / lead to unintended unwinds.

Pickup any tome of quantitative financial literature and the two foundational tenants underwriting it are as follows: risk is defined by standard deviation of price movement, and returns are distributed in a univariate fashion. These were necessary assumptions at the time when erecting a discipline known as financial economics. The field gave rise to covariance optimization and regression-based factor analysis, the foundations of active portfolio management. This ultimately led to VaR, helping financial markets along the way increasingly move to electronic implementation and *seamless transfers of risk*. However, this procession has not been without bumps along the way.

__A Trip Down Risk Management Lane__

Before humans developed a thorough understanding of statistics and its use in managing risk, we spent several centuries trying to measure risk/reward and to understand probability. If you’re interested in well-written histories of this, we’d recommend * Against the Gods* by Peter Bernstein and Stephen Stigler’s

__The History of Statistics__. I want to focus on two legendary mathematicians and some recent discourse surrounding their philosophy with the goal of unpacking some of the thorniest phenomenon as it pertains to abnormal risk events.

The first historical figure is Blaise Pascal, one of the finest minds of the 17^{th} century. Pascal had personal demons and was rankled by his pursuits in mathematics versus an unshakable attraction to faith. To help reconcile these divergent paths, he formulated a wager that sought to quantify the risk/reward of belief/disbelief. In short, Pascal’s wager surmised that if an outcome has an infinite outcome, one should always invest if the price is nominal. In context, if there is the slightest chance that God exists, you should believe in God, because if it were the case that *she* doesn’t exist, then your loss (*I guess going to Sunday school?*) is relatively benign compared to an eternity of damnation.

You can financialize this concept through the lens of options markets. If you can hedge yourself from permanent capital loss through options, it makes sense to pay a finite premium for that peace of mind. A reciprocal view would be venture capital, where losing a pre-determined finite amount is worth the wager if one can make a near infinite return on a modest wager. But does this view hold true to modern investing?

Our PnL is not the divinity (*as much as we may obsess over it*), and too many investors have been burned for too long hedging tail-risk, squandering away an ability to compound a superior rate of return and build a comfortable cushion for the periodic downturn in markets. Remember, markets historically trend up, and hedging is an expensive proposition unless it delivers truly asymmetric reward. Further, if one is allocated to various betas that are reflections of aggregate economic activity (e.g., an ETF tracking the Russell 3000), then a permanent loss of principal isn’t a straightforward matter. In the event that a broad economic instrument is truly a zero, no hedge will suffice save for guns, water, and twinkies.

Permanent capital loss for most institutions is a function of assuming exotic risk in search of alpha, or through the structure of their fiduciary obligations. It is less driven by the potential for core exposure to be permanently impaired. The role asset allocation models play in diversifying risk dampens the need for expensive catastrophe protection. Through diversification, one can effectively hedge without spending premium on loss-making contracts. Ultimately, investing became about adjusting risk and return for one’s fiduciary obligations. If you are a pension, your asset allocation is liability matched. If you are an endowment, it is though of as purchasing power + a giving quota. If you are an individual, it is most often built upon an expectation of retirement standard of living. Pascal’s wager feels a bit inadequate in dealing with the realities of these market dynamics.

The blogger / polymath / cryptoenthusiast Nick Szabo has written on this subject, coining a term *Pascal’s Scams*. He thinks that too often we price tail-events through the assumption of linear risk/reward: the expected value of a nearly impossible payout is equal to the payout potential times the (sometimes infinitesimal) probability. But as Nick notes, when we move to extreme tail scenarios, our understanding of probability is very poor, and so the premium for such a payout is often mispriced. Just look at the price of volatility before the spectacular pop in the first quarter of 2018. The reciprocal of course is the market maker who sells the insurance without truly understanding the value of the premium, and perverse incentives can make this disastrous (more on that later). Academics have aimed to update their models and understanding of risk to account for this nonlinearity. Bates (1991) wrote about the inadequacies of Black-Scholes (1973) in pricing options, due to jump risk and more broadly nonstationary behavior. Before Bates, we grappled with this phenomenon in a simpler construct.

__St Petersburg Paradox__

Our second historical figure is Nicolaus Bernoulli, a titan in his own right but somewhat overshadowed by the accomplishments of *other Bernoullii* of his day and age. Nicolaus came up with a thought experiment called The Saint Petersburg Paradox. It attempted to price the value of a game (successive coin-flips) where the aggregate upside was potentially infinite, but the value of each discreet event was fixed. The paradox helped ultimately unpack loss aversion principles and the shakiness of business models that accept premium but payout unlimited downside. Cue call AIG FP, circa 2006:

Do the old timers remember this doozy of a chart? I sure do. Every market cycle or so, we get a rude awakening to the siren’s song of ungodly returns with little realized volatility. The recurrent reminder has been that returns that are *too good to be true* may have risks that aren’t readily apparent due to the skewness of their distribution. It took a few burns from a hot stove for these investors to learn about this. LTCM is the granddaddy, but let us not forget Amaranth, Swiss-franc de-pegging and the 1Q18 vol-strike. Unlike in theoretical frameworks, markets have a recursive ability to exacerbate phenomenon. The lucrative act of betting on mispriced risk in the lead-up to the Great Financial Crisis caused an amplification of that very risk, creating greater skewness in the realized outcome than one could ever expect. Given this grave realization, risk practitioners have tried to come up with new metrics to deal with skewed distributions in assessing realized versus latent risk. Metrics such as the Johnson-Omega ratio took a crack at the inadequacies of mean-variance:

__Practical Applications?__

All of this is fine in its *mathiness*, but what does it mean about navigating portfolio risk today? For one, don’t be shocked when these bursts of volatility occur, because they will occur until a new generation of speculators take their turn burning their hands on the stove. Stive to construct portfolios that can weather them. For the lay investor who doesn’t invest in derivatives, doesn’t use leverage, and doesn’t think about mean-variance optimization, your best therapy is a long-term perspective. Aim small, and you’ll miss small. Time is your friend.

For those *cognoscenti* that use some degree of the above, being wise to the risk of gross (not net), being inherently skeptical of a VaR calculation, approaching optimization with skepticism, and understanding unlimited downside payoffs is mandatory. These risks, by being nonstationary and reflexive, require us to think more broadly about portfolio protection. A great mental model for doing so is a cross-disciplinary approach to risk management, rather than a singular focus on statistics or balance sheet decomposition.

Case in point, the principles of our modern economies (a cost of capital seeking an optimal return) and human behavior (incentives driving outcomes) leads us to *expect* non-linear risk to metastasize in complex systems that *inherently* hide risk. In 2008, it was the mispriced risk of subprime debt due to its ability to find refuge in complex securitization. In 1998 it was the imbalance of USD denominated debt in emerging market currencies that enjoyed weak underwriting standards due to the intoxicating appeal of their growth rates. In 2018, could it be Chinese corporate debt and shadow banking? We have several potentials for hidden macroeconomic risk that can pop up as the high tide of low interest rates and extended valuations wash away. This isn’t a paen to chicken little or an attack on scientific rigor. It’s a call to tread carefully, ye who feel safe with complexity.

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