How Outlier Hunters Exploit Convexity to Achieve Infinite Yield

How Trend Followers Exploit Convexity: The Art of Capturing Outliers In the world of finance, where smooth returns are often seen as the gold standard, trend followers and outlier hunters dare to embrace the chaos of markets. They thrive in uncertainty, leveraging volatility, diversification, and the principles of convexity to turn rare, transformative events into exponential growth. Introduction: Trend Following and Convexity In previous discussions, we explored the fundamental nature of convexity in financial markets (What is Convexity and Why It Matters) and how traditional risk models, such as those in Sharpe World, fail to capture the chaotic, nonlinear dynamics that define real-world markets. Building upon these principles, this article examines how trend followers—those who systematically hunt for market outliers—harness convexity to create resilient, high-potential portfolios. Trend following strategies are designed not to predict, but to react—to identify and capitalize on emerging trends. The key to their success lies in their ability to cut losses short, let profits run, and exploit asymmetry—a hallmark of convexity. While conventional portfolio managers seek smooth, linear returns, trend followers embrace volatility and the fat tails of return distributions, positioning themselves for rare, transformative market moves. Trend Following and Convexity: A Natural Fit Trend following and convexity are inherently aligned. Convexity is about asymmetry, ensuring that small losses are absorbed while massive gains are captured. This is precisely how trend followers operate: Small, controlled losses – A disciplined approach ensures losses are minimized and predefined. Unlimited upside potential – Once a trend is identified, trend followers let winners run, maximizing potential convex payoffs. Diversification – Trend followers operate across multiple, uncorrelated markets, ensuring they are always exposed to potential outlier events. Trend Following vs. Options: Convexity with Limits While trend following is a highly effective convex strategy, unlike options, it does not offer complete protection against all tail events. Short, sharp corrections – Trend-following models, particularly medium and long-term strategies, take time to adjust to new market conditions. When markets experience sudden reversals, trend-following models can get whipsawed, leading to a string of small losses before identifying a new trend direction. Longer-term corrections – When trends persist for an extended period, trend-following fully invests in convexity, riding out trends and maximizing returns from fat-tailed market events. Options provide immediate asymmetry, where downside risk is clearly defined, and upside is theoretically unlimited. Trend following, by contrast, requires active management and model adjustments, leading to some exposure to short-term market noise. Greater diversification potential – Unlike options, which have a relatively restricted set of choices in terms of strikes and expirations, trend-following portfolios typically comprise hundreds of uncorrelated return streams, spreading convexity exposure across a vast range of markets. No insurance cost – Options require a premium to provide tail-risk protection, effectively acting as an insurance cost. Trend following, on the other hand, does not have an equivalent premium cost, making it a cost-efficient way to harness convexity without needing to pay for insurance upfront. How Trend Followers Construct Convex Portfolios Trend followers integrate three key convexity principles into their portfolio construction: 1. Embracing Volatility and Fat Tails Traditional investing penalizes volatility, assuming it equates to risk. As we discussed in The Pitfalls of Sharpe World Thinking, this is a fundamental misunderstanding. Risk is not about historical volatility, but about future unpredictability. Trend following strategies thrive in volatility, recognizing that large price moves drive long-term compounding. The ability to stay in the game through disciplined risk management ensures participation in the fat-tailed events that drive exponential returns. Example: The Cocoa price explosion in 2024—a classic trend-following convexity play where patient traders who endured small losses were rewarded with an outlier move that transformed their portfolio performance. 2. Asymmetry: Small Losses, Big Gains Convexity is about tilting the risk-reward dynamic to one where the potential reward significantly outweighs the risk. Trend followers structure their strategies to capture this asymmetry: Predefined exit points – Losses are cut swiftly to avoid major drawdowns. Trailing stop mechanisms – Profitable trades are held as long as momentum persists. Path dependency is embraced – Unlike Sharpe World models that assume static market behavior, trend following recognizes that future price movements are influenced by prior trends. This approach ensures that a single outlier can offset dozens of small losses, driving long-term compounded wealth. 3. Diversification: Spreading the Convexity Net Wide Unlike options strategies that rely on a specific market outcome, trend following invests heavily in diversification, spreading exposure across multiple assets to capture convexity wherever it appears. Trading across multiple asset classes – Stocks, commodities, currencies, and bonds. Geographic diversification – Ensuring exposure across multiple economies and risk environments. Reducing correlation dependencies – Avoiding over-reliance on any single sector or region. This broad exposure enhances convexity-driven portfolio resilience, allowing for asymmetric return profiles across different market regimes. Trend Following as the Ultimate Convex Strategy In essence, trend following is a convexity-maximization strategy: ✅ Risk is predefined – No single trade can significantly damage the portfolio. ✅ Returns are uncapped – Trend followers allow market forces to dictate how far a trend can run. ✅ Survivability ensures participation – By keeping bet sizes small, trend followers remain in the game long enough to capture market outliers. Key Takeaways: The Convexity Advantage of Trend Following Markets are nonlinear and unpredictable – Trend following doesn’t fight this reality but capitalizes on it. Convexity ensures risk is asymmetric – Small losses are tolerated for the chance to capture extreme payoffs. Diversification amplifies convexity – Exposure to multiple markets increases the chance of catching outliers. Risk management is paramount – Staying in the game is more important than winning every trade. Conclusion: The Power of Convexity in Trend Following Traditional investing tries to smooth returns and eliminate volatility. Trend following embraces uncertainty, harnessing convexity to transform risk into opportunity. By applying convex principles—cutting losses short, letting profits run, maintaining small bets, and diversifying broadly—trend followers have built some of the most resilient and successful trading strategies in history. The path isn’t about predicting what will happen—it’s about positioning

The Pitfalls of “Sharpe World” Thinking: Why It Fails to Capture Convexity

The Pitfalls of “Sharpe World” Thinking “Sharpe World” thinking is inadequate for today’s complex and unpredictable markets. It fails to account for the chaotic, non-linear realities of financial markets. Introduction: The Pitfalls of “Sharpe World” In modern finance, the pursuit of risk-adjusted returns has led to a widespread reliance on measures like the Sharpe ratio, standard deviation, and Value at Risk (VaR). This framework—what David Dredge aptly calls ‘Sharpe World’—has shaped how investors think about risk, yet it is fundamentally flawed. The 2008 financial crisis wiped out nearly $19 trillion in global wealth, yet just months before, risk models showed no indication of an impending collapse. Why? Because they relied on the past to predict the future—ignoring the fundamental reality that the most significant market moves are always unexpected. The regularity of unforeseen market crises that blindside investors is not an anomaly but a direct consequence of this flawed framework. Risk models work until they catastrophically fail, reinforcing a dangerous illusion of control over markets that are inherently unpredictable. The fundamental issue is that risk models tend to measure what has already happened, rather than preparing for what has never occurred before. When the next major crisis arrives, it will not be a variation of past events—it will be something completely different. Refer to the table below that lists some of the most significant financial crashes that blindsided the investment community despite their reliance on traditional risk models. Risk is Not Volatility One of the greatest misconceptions in finance is the belief that risk equates to volatility. This assumption underpins much of Modern Portfolio Theory (MPT), the Efficient Market Hypothesis (EMH), and standard risk management practices. But history has shown us that the most damaging risk events are not the ones we expect, but the ones we don’t see coming. If risk could be forecasted using historical distributions, then risk management would be easy. But every major financial crisis—from Black Monday to the Global Financial Crisis—occurred because the models failed to anticipate the unexpected. As Dredge, describing Nassim Taleb’s insights, puts it: “Understanding is a poor substitute for convexity.” Risk isn’t about predicting the next crisis; it’s about building a portfolio that can survive and exploit uncertainty. The Flawed Assumptions of ‘Sharpe World’ Traditional financial models rest on several assumptions that, while convenient for theoretical frameworks, have repeatedly failed in real-world market conditions. These flawed assumptions create a false sense of security, leading to systemic fragility and underpreparedness for extreme events. Risk can be quantified using historical data. Why it’s flawed: Risk is not static—it evolves dynamically. Using past data assumes that future risks will mirror historical occurrences, but market crises often stem from unprecedented shocks. Models built on past distributions fail to account for the unpredictability of future events. Implication: Investors relying on historical risk metrics are often blindsided when markets deviate from historical norms, leading to severe miscalculations in risk exposure. Markets behave in a linear fashion, following normal distributions. Why it’s flawed: Real-world markets are nonlinear and exhibit fat-tailed distributions, where extreme moves happen more frequently than predicted by normal distributions. Standard financial models underestimate the probability and impact of rare, high-magnitude events. Implication: Risk management strategies based on linearity fail to anticipate market dislocations, exposing portfolios to devastating tail risks. Correlations persist into the future. Why it’s flawed: Correlations are highly unstable and tend to shift dramatically during periods of market stress. Assets that appear uncorrelated in normal times often become highly correlated in crises, negating diversification benefits. Implication: Portfolio designs that assume stable correlations fail when they are needed most, leading to simultaneous losses across supposedly diversified holdings. Expected returns can be estimated with confidence. Why it’s flawed: Markets do not follow predictable patterns, and expected returns fluctuate based on shifting macroeconomic conditions, sentiment, and structural changes. Using historical averages to project future performance ignores the reality of ever-changing market regimes. Implication: Investors who rely on projected returns may over-leverage during favorable periods and under-allocate when opportunity arises, leading to suboptimal compounding over time. All volatility is bad and should be minimized. Why it’s flawed: Not all volatility is detrimental. While downside volatility can be damaging, upside volatility represents opportunity. Attempts to smooth returns by suppressing volatility often reduce exposure to large, outsized gains, capping long-term compounding. Implication: Strategies focused on reducing volatility at all costs often sacrifice convexity, failing to capitalize on beneficial market trends while remaining overly exposed to unseen risks. These incorrect assumptions lead to a fragile approach to risk management—one that attempts to control and predict risk rather than adapt to and exploit it. Markets are inherently nonlinear, unstable, and shaped by extreme events, yet Sharpe World thinking forces investors into models that are precise in theory but dangerously inaccurate in practice. The failure to recognize these flaws leaves investors exposed to sudden shocks and unprepared for the rare but defining market moments that drive long-term performance. The failure to account for fat-tailed events and changing market regimes means that many investment strategies operate without effective brakes. Investors unknowingly rely on historical relationships to contain risk, assuming that what worked in the past will work in the future. But without proper braking mechanisms, these strategies leave investors fully exposed when markets take an unexpected turn. Investment Strategies Without Brakes: A Recipe for Disaster A core problem with many investment strategies is that they lack brakes—mechanisms that prevent catastrophic drawdowns and allow for adaptability in volatile environments. Without brakes, investors are fully exposed to market downturns and rely on historical relationships that may not hold up in the future. Some examples of strategies without brakes include: The 60/40 Portfolio: Relies on historical negative correlation between stocks and bonds to mitigate risk. However, in times of rising inflation or systemic crises, both asset classes can decline simultaneously, removing any perceived protection. Buy and Hold (Long-Only) Portfolio: Has no mechanism for managing downside risk. Investors are fully exposed to extended drawdowns, hoping for a long-term recovery that may take decades. Mean-Reverting Strategies: Assume

What is Convexity and Why It Matters for Trend Following

What is Convexity and Why It Matters Convexity explains how small changes in certain environments can lead to disproportionately large outcomes. Introduction Financial markets are anything but predictable. Despite the desire for smooth, steady returns, markets inherently exhibit nonlinearity, unpredictability, and fat-tailed distributions. This reality demands a strategy that embraces uncertainty rather than one that attempts to suppress it. The key to thriving in this environment lies in convexity—a principle that transforms volatility into opportunity through asymmetry, dynamic risk management, and compounding power. What is Convexity? Convexity is the concept that small inputs can lead to disproportionately large outcomes—both positively and negatively. Unlike linear relationships, where returns grow proportionally to risk, convexity introduces a curved dynamic, where favorable volatility accelerates gains while unfavorable volatility applies brakes. Markets themselves exhibit convexity, meaning that traders who attempt to force smooth, linear returns onto an inherently wiggly world are destined for failure. All equity curves ultimately reveal either a convex or concave signature, depending on how they respond to market uncertainty. Convex portfolios (positive skew) embrace asymmetry, keeping losses small while allowing for large, outsized gains. Concave portfolios (negative skew) suppress volatility and attempt to smooth returns but ultimately suffer from large, catastrophic drawdowns. Most traders unknowingly operate within a concave framework, where the illusion of stability comes at the cost of hidden risk. Convexity, on the other hand, transforms the frown of concavity into the smile of opportunity, ensuring that portfolios are positioned to benefit from market dislocations rather than be blindsided by them. Convexity and Skew: The Essential Distinction The hallmark of a convex portfolio is positive skew, while a concave portfolio is characterized by negative skew. Positive Skew (Convexity): Frequent small losses, punctuated by rare but disproportionately large gains. This is seen in trend-following, long-volatility strategies, and asymmetric portfolio structures. Negative Skew (Concavity): Frequent small gains, but with occasional, devastating losses. This is typical of mean-reversion strategies, short-volatility positions, and leveraged martingale models. Because markets are inherently nonlinear and fat-tailed, every strategy will eventually reveal a convex or concave profile. Convex strategies thrive by exploiting uncertainty, while concave strategies eventually collapse under its weight. The Goal of Convexity: Optimizing Compounding Many investors fall into the trap of targeting an optimal average speed in a market environment that is anything but smooth. A prime example is the S&P 500, which has an average return of 8% per year—but this average obscures extreme variability: In some years, returns exceed 20%. In crisis years, losses exceed 30%. Attempting to target the average leads to dangerous missteps: Leverage increases exposure during downturns, compounding losses. Profits are taken prematurely in favorable regimes, capping upside. This is akin to a racecar driver maintaining the same speed on all parts of a winding track. Without the ability to brake on sharp turns and accelerate on straights, the driver will either crash or fail to compete effectively. Why Convexity Wins the Compounding Race Convexity prioritizes risk-adjusted adaptability rather than forcing an artificial smoothness onto a chaotic market. The convex trader slows down when uncertainty rises and accelerates when conditions become favorable, creating an optimal trajectory for long-term compounding. Braking (Risk Mitigation): Avoids devastating losses by cutting risks during adverse regimes. Acceleration (Opportunity Capture): Capitalizes on major market trends and dislocations. Non-Predictive Adaptability: Adjusts dynamically rather than relying on fragile forecasts. Those who embrace convexity understand that attempting to force stability in an unstable world is a losing battle. Instead, they design portfolios that thrive on adaptation, asymmetry, and compounding, ensuring that when opportunity arises, they are positioned not just to participate, but to dominate the market landscape.  

Understanding Warehoused Risk and Why Stops are Critical Risk Management Tools for Classic Trend Followers

Understanding Warehoused Risk and Why Stops are Critical Risk Management Tools for Classic Trend Followers This blog explores the concept of warehoused risk and how effective portfolio management can enhance returns while mitigating risk. When trading, we often focus on individual strategies and their performance metrics, such as drawdown (DD) and compound annual growth rate (CAGR). However, the real power of trading lies in managing a portfolio of these strategies. This blog explores the concept of warehoused risk and how effective portfolio management can enhance returns while mitigating risk. Importantly, it demonstrates a curious conservation law in investing: “Risk cannot be eliminated from a portfolio; it can only be transferred within it.” The only way to release risk from a portfolio is by eliminating the risk contribution of an individual return stream by exiting that position. We will demonstrate why stops, though often criticized as inefficient for mitigating risk, are essential at the portfolio level. They provide risk release valves to mitigate current risk and allow the portfolio to absorb new risk in the future. Not having stops in place can detrimentally affect portfolio performance, especially when market conditions arise that have never been seen before in backtests. These new, unforeseen environments can expose the warehoused risk that exists in a portfolio, which may not have been previously observed. Single Strategy vs. Multiple Strategies Consider a single trading strategy on a single market. This strategy might produce a return profile with a 20% drawdown and a 7% CAGR. If we multiply this strategy five times, trading five identical systems on the same market with the same allocation, the drawdown increases to 100%. This happens because the drawdown in each system occurs simultaneously, resulting in a linear relationship between leverage and drawdowns. Multiplying the position size by 5x causes the drawdown to increase fivefold. However, the relationship with CAGR is not linear. While multiplying the strategy 5x might achieve a drawdown of 100%, the corresponding CAGR might only increase to around 30%. This is because CAGR is a path-dependent and nonlinear metric, where increased volatility suppresses compound growth. Now, what happens if we diversify our approach? Suppose we develop 10 different, uniquely configured trend-following (TF) strategies for the same market. Each of these strategies has a 7% CAGR and a 20% drawdown occurring at different points in time. The portfolio of these 10 strategies might then produce a CAGR of 40% but with a drawdown of only 40%. This is because the risks are spread out due to the lack of perfect correlation between the strategies. The drawdowns of each unique strategy do not coincide, and each return stream offers correlation offsets at different points in time across the entire time series. Furthermore, the CAGR is increased as the volatility drag associated with the drawdown of the entire ensemble at 40% is far less than the alternative of 100%. By diversifying, we distribute risk across various strategies, each contributing to the overall performance at different times. This risk spreading, due to the lack of perfect correlation, allows for a more stable and robust portfolio, enhancing returns while managing drawdowns more effectively. The Principle of Warehoused Risk Warehoused risk is a crucial concept in portfolio management. It represents the total risk present in all return streams, calculated as if each return stream went to zero at a specific point in time. By summing the potential risk contributions of each return stream, we arrive at the total risk summation, known as the warehoused risk of that portfolio at a given moment. This theoretical limit adheres to the conservation law that risk can never be eliminated, only transferred within the portfolio. Another term used to describe warehoused risk is “Portfolio Heat.” Many investors overlook this potential risk lurking in their portfolios—the risk of total collapse if all the risk held by a portfolio is suddenly released at once. Think of warehoused risk as akin to a “risk sponge.” The more we diversify and add new return streams to a portfolio, each with the same risk contribution, the more we pack warehoused risk into the portfolio. Despite the presence of warehoused risk, the risk investors typically pay attention to are measures such as the Sharpe ratio, Sortino Ratio, MAR (maximum drawdown), Ulcer Index, and other risk metrics. These measures are always far lower than the warehoused risk that actually resides in a portfolio. They reflect how individual risks within a portfolio offset each other and how risk events are dispersed across the time series of different return streams. However, these metrics often understate the actual risk potential within a portfolio. These risk measures typically assess the volatility of portfolio returns over time, a consequence of how discrete return streams interact. Some risks cancel each other out, resulting in a net portfolio variance measure, such as standard deviation or maximum drawdown, occurring at specific points in time. However, these proxy measures understate the possible risks if a new market regime emerges—one that has never been experienced in backtests and significantly alters these risk metrics, reflecting the higher warehoused risk inherent in the portfolio. Understanding warehoused risk helps investors recognize the potential hidden dangers within their portfolios. By acknowledging this risk and managing it through diversification and strategic use of stops, we can create more resilient portfolios that are better prepared for unexpected market conditions. The Role of Stops in Portfolio Management Trend followers often use stops as a critical tool for managing portfolio risk. Stops should be seen as risk release valves for the entire portfolio, preventing warehoused risk from becoming overwhelming due to any contributing return stream in unfavourable market conditions. While some traders argue that stops detract from performance compared to other exit measures, in portfolio management, stops are crucial for maintaining the positive skew of the entire collection of return streams and managing total portfolio heat. Consider this: in a portfolio comprising potentially thousands of return streams, there is always the possibility that many return streams could suddenly become positively correlated, potentially

Navigating the Complexities of Risk in Trend

Navigating the Complexities of Risk in Trend In a Non-Linear World View Straight Lines with Caution: Risk Management Principles for Trend Following The ability to adeptly navigate risk stands as a crucial skill for both investors and traders. The quest for robust investment strategies has traditionally leaned on a set of established risk metrics, with the Sharpe Ratio, Sortino Ratio, and Standard Deviation at the forefront. These metrics have provided a foundational framework for assessing the risk-adjusted performance of various investment approaches, offering a semblance of predictability and control in the inherently unpredictable nature of financial markets. The Sharpe Ratio, for instance, has been widely revered for its simplicity and effectiveness in conveying the amount of excess return per unit of risk, with risk quantified as the standard deviation of returns. Its appeal lies in its straightforwardness, allowing for a quick comparison of different investment opportunities under a common risk-return lens. Similarly, the Sortino Ratio refines this concept by focusing solely on downside risk, aligning more closely with the typical investor’s aversion to losses. Standard Deviation, on the other hand, offers a direct measure of volatility, serving as a proxy for the uncertainty inherent in investment returns. Despite their widespread adoption and inherent virtues, these traditional metrics are not without limitations, especially when applied to the intricate domain of trend-following strategies. Trend-following models, characterized by their reliance on capturing sustained directional market movements, embody a unique set of attributes that challenge the applicability of conventional risk assessments. The core of these strategies lies in their path-dependent nature, where the sequence and timing of market trends significantly influence their performance outcomes. The primary shortfall of metrics like the Sharpe and Sortino Ratios in this context is their inherent assumption of symmetry and normality in return distributions. These metrics do not distinguish between upward and downward volatility, treating all fluctuations around the mean as equal contributors to risk (Refer to Figure 1). This oversimplification glosses over the nuanced dynamics of trend-following strategies, where the asymmetry of returns—frequent small losses punctuated by occasional large gains—is a defining characteristic. Moreover, the path-dependent nature of trend-following strategies introduces a layer of complexity that traditional metrics are ill-equipped to handle. The success of these strategies hinges not just on the magnitude of market movements, but on the sequence in which these movements occur. A series of small gains followed by a significant upward trend can result in a vastly different outcome than the same trend occurring in reverse order. This aspect of path dependence is critical in understanding the risk and potential of trend-following models, yet it remains conspicuously absent from the risk assessment toolkit provided by traditional metrics. Figure 1: Three Different Strategies with Three distinct Paths of Returns with Identical Sharpe Ratios and Standard Deviations. Without seeing the NAV chart and only viewing the classic risk-return metrics, an investor would be indifferent: all three strategies would look equally good, however the impact of these paths on a return series have serious consequences for compounded wealth. In essence, while traditional risk metrics like the Sharpe Ratio, Sortino Ratio, and Standard Deviation have served as valuable tools in the arsenal of investors and traders, their application to trend-following models reveals inherent limitations. The dynamic and complex nature of these strategies, thriving on the ebbs and flows of market trends, demands a more nuanced approach to risk evaluation—one that considers the asymmetry of returns and the pivotal role of path dependence in shaping investment outcomes. The Limitations of Conventional Risk Metrics The limitations inherent in conventional risk metrics such as the Sharpe and Sortino Ratios extend beyond their mathematical formulations to the very core of how we perceive and measure investment risk. These metrics, while elegant in their simplicity, often fall short of providing a holistic view of an investment’s risk profile, especially in the context of specialized strategies like trend-following. The Sharpe Ratio, for instance, has been a linchpin in the arsenal of risk assessment tools, offering a succinct measure of risk-adjusted performance. By dividing the excess return of an investment by its volatility, it ostensibly provides a clear indicator of the return an investor can expect per unit of risk undertaken. However, this metric’s reliance on standard deviation as a proxy for risk introduces a critical blind spot: it does not differentiate between positive and negative volatility. In the realm of trend-following strategies, where profits often stem from “riding” prolonged market trends, this failure to distinguish between beneficial volatility (upside) and harmful volatility (downside) can lead to misleading interpretations of an investment’s true risk profile (Refer to Figure 2). Figure 2: The geometry, or the paths of returns, have significance for compounded returns. Contrary to popular opinion there are better geometries for compounded wealth apart from straight lines. Given that Sharpe ratios penalise beneficial volatility, we observe that Example 1 (the straighter line) has a far higher Sharpe than Example 2 but a far lower CAGR. The direction of the volatility is crucial for wealth generation which can be observed when comparing the terminal wealth of strategies with negative skew and positive skew. Moreover, the Sharpe Ratio’s implicit assumption of normally distributed returns does not hold water in the unpredictable seas of financial markets, where extreme events (often referred to as “black swan” events) are not as rare as traditional models would suggest. This discrepancy becomes even more pronounced in trend-following strategies, which, by design, aim to exploit these very outliers—large, sustained market moves—rendering the Sharpe Ratio’s insights less applicable, if not entirely moot, in evaluating such strategies. On the other hand, the Sortino Ratio, often touted as an improvement over the Sharpe Ratio, narrows its focus to downside volatility, ostensibly aligning more closely with investors’ natural aversion to losses. By considering only the negative deviations from the mean return, the Sortino Ratio aims to provide a more relevant measure of “bad” risk. While this adjustment marks a step towards a more nuanced understanding of risk, it, too, is not without its shortcomings. Specifically,