Be Aware, Adapt, Innovate--or What We Learn from Mandelbrot, Noah, and Covid-19

Reflections on Risk and Investment Management

Risk Categorization and the Joseph and Noah Effects

Just about a year before the breakout of COVID-19, I published a short article at GARP titled “In Risk Management, Fear is a Step to Investment Wisdom,” which discusses how to transform investment uncertainty and risks into smarter decisions and more resilient outcomes by using extremes-aware modeling approaches. I find this topic even more relevant today, when uncertainty spreads much farther than investment outcomes, and while the market is facing unprecedented times.

In brief, the article classified financial risks into three categories by their severity, transmission, and impact on the overall system and offered families of statistical models as well as risk and portfolio construction high-level practices to anticipate and cope with respective environments.

Table 1 /excerpt from https://www.garp.org/?cx=016219603818583579766%3Alny-8m_ldps&q=FactSet/#!/risk-intelligence/market/investment-management/a1Z1W000004BTh9UAG /

The first category of risks and associated models assume persistency. In other words, things tend to stay the way they've been behaving (or manifesting) recently. Some examples:

• Successful people tend to stay successful;
• Winning teams tend to keep winning; and,
• Technologies exhibiting fast growth for the last year have a high probability of fast growth next month.

Mandelbrot—the father of the theory of roughness and one of the most renowned mathematicians of recent times whose work has profound implications for the world of finance [1], [2],—named the phenomena governing this first category of risks and associated environment “The Joseph Effect,” after the biblical story of how Joseph interpreted the Pharaoh's dream of seven fat cows and seven gaunt cows to mean that there would be seven prosperous years followed by seven lean ones. This is the “persistency pattern.” Models that only capture the Joseph effect lead to myopic behaviors that presume that the current times may continue undisrupted for a long time. There is also the Noah Effect, as named by Mandelbrot, after the story of the Great Flood, which describes discontinuity. This effect describes the phenomena that when something changes, it can do so abruptly. For example, a stock share priced at $35 can quickly fall to $5 without ever being priced at any intermediate level if something significant triggers its collapse.

The Noah and Joseph Effects describe two opposing phenomena of nature: trends exist and last, but they can vanish as quickly as they come.

Referencing my previous article, the Noah Effect accounts for Tails Risk and Structure breaks. The probabilistic modeling paradigm discussions in academia have started as early as the beginning of the century [6], and continue to be some of the most exciting areas of quantitative mathematics [3], [7]. Some may argue that the Tails Risks as described can be modeled by a Levy process with infinite intensity, while Structural breaks can be represented in the form of compound Poisson processes (finite intensity Levy processes). My opinion is that in the world of finance both occur in the form of the prior, however, with varying degrees of fat-tailedness. In any event, this question is of secondary importance.

What is however important is that for periods where events unfold based on the Noah paradigm, what matters is to “be aware, adapt, and innovate.”

FactSet Fat-Tail model DNA incorporates both of those major phenomena. It operates under the assumption of continuous awareness of extremes. The model has an additional time-varying characteristic that accounts for the “local-in-time” probability of extreme events. It can be visualized by what we call Fat-Tail Indicator. This indicator, bundled with Vol level can provide insight into the local market regime and build “awareness” on upcoming extremes:

Table 2: Regimes

This series will go through approaches on managing risk and supporting asset-allocation and portfolio construction during periods of Tail Events and Structural Breaks and will illustrate the concepts through the lens and the current time-window of the COVID-19 crisis. In particular, we will focus on:

1. Awareness: Fat-Tailed Indicator, Value-at-Risk, and Expected Tail Risk measurement. Interpretation for short horizons and its implications for investment management.
2. Adaptation: Risk assessment at the mid-term horizon and reconstructing the tactical asset allocation mix.
3. Adaptation: Stress-testing and quantitative assessment of the probability of subjective stress scenarios.
4. Innovation: Multi-period optimization for long-term allocations incorporating stress-tests and investment objectives.
5. Innovation: Minimum Tail Risk Smart-Beta.

Throughout the entire discussion, we will focus on illustrating the importance of capturing all the above-described phenomena and the value that such model awareness brings to the investment performance.

The Extremes-Awareness Translated to Risk Management

The markets’ pre-disposition to extreme events is dynamic and changes based on various factors. It is vital to build an objective criterion measuring the level of such pre-disposition through time. This is what we call a Fat-Tail indicator. It is derived from the concepts of Expected Tail Loss (ETL)—the average loss when the loss is greater than a selected Value-at-Risk Threshold, typically viewed at 99% Confidence Level and its upside counterpart—Expected Tail Return (ETR), which represents the average gain if the gain is higher than the maximum possible gain achievable with a certain small probability (typically 1%).

Specifically, the Fat-Tail indicator is a function of ETL and ETR and represents both tail-fatness and skewness:

It captures both the excess probability of extreme losses via the difference of the Fat-Tail ETL and the ETL under the Normality assumption and the asymmetry (i.e., whether the extremes are more probable on the downside rather than on the upside) via the difference between the Expected Tail Loss and Expected Tail Return Fat-Tailed versus Normal spreads.

The dynamics of the Fat-Tail Indicator for MSCI US provides a valuable insight into the different market regimes and transitions.

In Figure 1, we depict the following for the period January 1, 2018—Маrch 6, 2020 (from the top to the bottom subplots): the cumulative performance of the U.S. stock market; stock market returns together with the Normal and FactSet Fat-Tail model 99% VaR measures; stock market return standard deviation, and the FactSet Fat-Tail Indicator for the U.S. equity market.

Three observations are worth noticing. First, the recent market downturn has (unsurprisingly) led to a spike in the realized return volatility while the Fat-Tail Indicator is sustained at its already elevated level. Second, as discussed above, the FactSet Fat-Tail VaR has a statistically strong predictive power with 11 exceedances YTD and one exceedance since January 1, 2018. Third, and more interesting, the Fat-Tail Indicator has sustained elevated levels (above 50 percent) for almost the entirety of the last two years. The model has therefore been gradually “pricing in” increasing turbulence via individual news (i.e., trade deal negotiations, political developments, etc.) and market and economic disturbances that have been contributing to the build-up of “turbulence pressure”—a phenomenon that is not apparent through the lens of the traditional volatility metrics. This is most clear in the second half of 2019 when the Fat-Tail Indicator records a significant increase and then maintains that elevated level prior to the spike in realized volatility in the subsequent equity market downturn of early 2020. That way, the model was already operating at an elevated expectancy of extremes and thus provided a significantly better real-time assessment of the underlying market risk compared to more traditional risk measures during the COVID-19-triggered financial markets collapse. The period at the beginning of the year corresponds to a False-safety period/Pre-stress period as defined in Table 2; the period of the first three weeks of March—to a Developing Crisis period; and the last week of March—to a Developed Crisis period where we see the Fat-Tail Indicator starting to go down despite the extreme values of Vol.

Figure 1: MSCI USA Performance, Volatility and Fat-Tail Indicator

Components of Extremes-Aware Model and the Nature of Extremes

A reasonable question of how to make the model aware of market discontinuities arises.

We can attempt to explain it via some observations on the dynamic of the current pandemic and contrasting it to normal conditions. During an epidemic, the number of people that fall ill over a certain unit of time gets disproportionally large, multiples of times larger than during normal times. When the epidemic transmits across territories, it turns into a pandemic, i.e., the increase in the number of contaminated individuals transmits from one place to another very quickly. If we define a time measure of an “infection-day” for a given virus as the physical time for which a given number of individuals, for example, 24, become infected (with each individual being considered a “new event” or “news” which marks the virus hour), instead of using the number of hours, then an epidemic state is a period when the disease’s “clock” runs much faster than our physical time. Moreover, it becomes highly “random” and speeding-up with the development of the epidemic state. The only thing we could try to do is to “slow-down” the time-speed of the disease via isolation, i.e., to “flatten the curve” as the hospitals can’t absorb what happens within a unit of physical time.

For modeling other types of phenomena, we may need to choose the “time measure” accordingly, depending on the situation—instead of the physical clock tick, it could be “specific number of raindrops,” or “the sell orders,” for example.

Translating this to financial markets, we can say that the market is in an “epidemic” or stress regime when a large number of important events or news happens much faster than during normal market regimes. Specifically, the “market clock” intensifies relative to our physical time and the returns of a given market need to “absorb” and reflect much more within a unit of physical time, which causes a crash. Here, “news” can be any important information: large trades or rapid trading, policy announcements, shifts in macro dynamics or sentiment, or outside factors. The intensity of the market time is an additional factor of a special kind (volatility-multiplicative), by definition ignored by the traditional models as they operate under the implicit assumption that the market clock always ticks in synchronicity with the physical time. The FactSet Fat-Tail model uses a so-called time-subordinated model to incorporate the market-clock intensity [4], [5]. Its dynamics can be extracted from the market returns using statistical approaches [3], [6], and are then simulated so that each main risk driver or market is attached to its market-time intensity.

In finance, we have different means to flatten the curve—we can’t slow down the market time per se, we can only try to make the new events less prominent by releasing counter-events—i.e., the monetary policy and related interventions. Note that such actions do not affect the Fat-Tail Indicator by reducing the difference between the Normal and the Fat-Tail ETL but via changing the skewness by pumping up positive “events.”

The second important property is the probability of the stress to transmit to several markets (i.e., to have the epidemic becoming pandemic)—the tail-dependence. Tail-dependence can’t be captured via the regular correlation concept as it only operates properly under normal conditions. We need to find a way to make the market time “clocks” of the different markets interconnected. Mathematically this could be achieved either via a non-gaussian copula function or via a multivariate time-subordinated process [3], [5].

What Does This Mean for Investment Outcomes?

We will illustrate it here via a limited example at a mid-term investment horizon (one month) by backtesting two optimization approaches for finding optimal tactical allocation mix. The rest of the discussion papers in this series would offer additional insight across asset-classes and investment horizons.

We contrast Mean-Variance and Fat-Tail ETL optimization approaches across three equity indices, five Fixed Income, and four commodity exposures for the period from December 31, 2002, through March 31, 2020, with monthly rebalancing, 10% turnover, and constraints as listed in Table 3:

Table 3: Asset Classes Optimization Constraints

The tail-aware asset-allocation optimal composition achieves better cumulative returns, lower drawdown, and higher risk-adjusted performance characteristics (see Table 4 and Figure 2). As we shall see in the subsequent parts of the series with greater details, this outperformance is due to (1) using a more adequate risk measure and (2) assessing the risk via a model that incorporates both market time intensity and tail dependence[1].

Figure 2: Historical performance of the Fat-tail Min ETL and Normal-Min Variance strategies

The Bright Side of the Noah Story

Neither this article nor Mandelbrot aims to say these periods of high-tail fatness and intensity of extremes predict the end of the world as we know it negatively. There is a positive side to the Noah story. It is a story of adaptation through efficiency—take only what has real potential to deliver value; Innovation—find the business means to transfer you to the new state; and perseverance—be able to wait “40 days.” A prerequisite to this is to be aware of possible extremes and take measures to prepare. FactSet models provide at least the “timber logs” to make your investment boat secure. A bit more precisely, they offer you the warning signal to start preparing now. As hinted here, and as we shall see more thoroughly in later parts of the series, FactSet Optimization could offer the means to adapt by finding a more suitable and resilient portfolio composition or asset-allocation mix.

The rest of the game plan—which is the “persevere and innovate” element, is a manifestation of what psychology calls eudaimonic well-being. The current health crisis and the economic disruptions certainly mark this period as a time of deep worry, grief, and stress. Our hedonic well-being has been disrupted as never before in recent history. Yet, it is exactly the time when vivid approaches to innovation and efficiencies can evolve, where human potential can flourish, and when we can experience this rarely understood eudaimonic excitement and satisfaction. Ironically or not, it is precisely the type of eudaimonic well-being that sets positively our immune system to fight infection and to develop the antibodies that are essential to survival, endurance, and thriving [8].

Through the rest of the series, we will go deeper into the quantitative models’ innovation aspects that FactSet’s Risk and Quant Group is working on and into how those relate to the current crisis. Certainly, innovation and perseverance go much beyond that, but we hope that we can offer a small part of it.

Next in the Series:

Part 1: VaR 99% and Covid-19:  Back-testing studies on the FactSet Short-term Risk model covering the period of Covid-19 Outbreak.
By Dr. B. Racheva-Iotova, Ivan Mitov PhD, Viviana Vieli, Velislav Bodurov

Part 2: Risk assessment at mid-term horizon and reconstructing the tactical asset allocation mix
By Ivan Mitov PhD, Velislav Bodurov, Ian J. Hissey

Part 3: Stress-testing and quantitative assessment of the probability of subjective stress scenarios
By Georgi Mitov PhD, Emil Margaritov PhD, Shamin Parikh, Dr. B. Racheva-Iotova

Part 4: Innovation: Multi-period optimization for long-term allocations incorporating stress-tests and investment objectives
By Georgi Mitov PhD, Ivo Stefanov, Chen Sui

Part 5: Minimum Tail Risk Smart-Beta
By Georgi Mitov PhD, Nikolay Radev, James Egginton

References

[1] B. Mandelbrot. The variation of some other speculative prices. The Journal of Business, 1967.

[2] B. Mandelbrot, R. Hudson. The Misbehavior of Markets: A Fractal View of Financial Turbulence. Basic Books, 2006; ISBN: 978-0465043576

[3] B. Güner, B. Racheva-Iotova, I. Mitov. Fat-tailed models for risk estimation. In Encyclopedia of Financial Models III, pages 731–755. Wiley, 2013; ISBN: 978-1-118-00673-3

[4] B. Racheva-Iotova Et Al. FactSet Fat-Tail Multi-Asset Class Model, Introducing Turbulence Adjusted Risk. White-paper, 2018

[5] B. Racheva-Iotova Et Al. FactSet Fat-Tail Multi-Asset Class Model, Theoretical Foundations. White-paper, 2018

[6] S. T. Rachev, S. Mittnik. Stable Paretian Models in Finance, WILEY, 2000; ISBN: 978-0-471-95314-2

[7] Wim Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, WILEY 2003, ISBN-13: 978-0470851562

 [8] B. Fredrickson, A functional genomic perspective on human well-being, Proc Natl Acad Sci U S A. 2013 Aug 13; 110(33): 13684–13689. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3746929/#r57

[1] FactSet MAC optimizer is expected to be released in June 2020.