Why the number the industry quotes you is mathematically guaranteed to overstate your wealth
In 2000, a 58-year-old teacher in Ohio invested her retirement savings in a diversified portfolio of mutual funds. Her financial adviser showed her a projection: at an average annual return of 8%, she would have enough to retire comfortably at 65. The projection was not wrong. The average return her funds delivered over the next thirteen years was, in fact, close to 8% per year. But when she turned 65, her portfolio was worth considerably less than the projection had promised. She had done everything right: she had stayed invested, she had not panicked during the crashes of 2001 or 2008, she had kept paying in. The projection failed her not through dishonesty but through a mathematical error that is almost never explained. The number her adviser used, the arithmetic average, is not the number that governs how wealth actually grows. There is another number, quieter and more consequential, that the industry almost never quotes. This series is about that number.
The question feels like it should have a clean answer. Ten percent per year. $100,000 starting capital. Compound it forward for twenty-six years. Your financial planner would show you $1,191,818. The fund’s marketing materials would show you $1,191,818. Every retirement calculator on the internet would show you $1,191,818.
The number is mathematically valid and practically meaningless. It describes a world that does not exist: one in which the return arrives smoothly, identically, year after year, as though delivered by direct deposit. In the world you actually inhabit, returns arrive violently. They arrive in bursts and droughts, in crashes and recoveries, in sequences that matter enormously for the money you end up with but not at all for the average that is printed on the page.
The industry quotes you one number. Compounding delivers another. The gap between them is the most important force in wealth creation that most investors will never be shown.
The Two Averages
There are two ways to compute an average return, and they give different answers. The difference between them is the single most consequential concept in long-term wealth creation.
The arithmetic average is the one you know. Add up all the periodic returns, divide by the number of periods. If a portfolio rises 20% in year one and falls 10% in year two, the arithmetic average is (+20% + −10%) / 2 = +5% per year. Simple, intuitive, and ubiquitous. It appears on every fund fact sheet, every pension projection, and every financial plan you have ever seen.
The geometric average is the one that determines your actual wealth. It answers the question: what single, constant rate of return, applied each period, would produce the same terminal value as the actual sequence of returns? In the same example, $100 grows by 20% to $120, then falls by 10% to $108. The cumulative gain is 8% over two years, which corresponds to a geometric average of roughly 3.92% per year. Not 5%.
The arithmetic average says 5%. The geometric average says 3.92%. And this is not a marginal discrepancy. It is a structural feature of how multiplication works, distinguishable from addition, and it scales relentlessly with time.
To see why the two averages differ, consider what happens mathematically. Addition is symmetric: +10 followed by −10 returns you to zero. Multiplication is not: $100 × 1.10 × 0.90 = $99, not $100. The loss of a given percentage always destroys more than the gain of the same percentage creates. This asymmetry is not a quirk. It is the defining characteristic of multiplicative processes, and investing is a multiplicative process. Your wealth tomorrow is your wealth today multiplied by (1 + today’s return). Not added to. Multiplied.
Every time you see an arithmetic average return quoted, you are looking at a number that overstates what compounding will actually deliver. The overstatement is small when volatility is low. It becomes enormous when volatility is high. And it has a name.
The Volatility Tax
The relationship between the arithmetic average and the geometric average is governed by a formula that should be printed on every fund fact sheet and never is:
Geometric Return ≈ Arithmetic Return – ½σ²
Where σ² is the variance of returns. Read it plainly: volatility is not a neutral measure of uncertainty. It is a direct tax on compounding. The higher the volatility, the wider the gap between the return the industry quotes you and the return your money actually earns. The tax is always positive. It always reduces geometric returns below arithmetic returns. And it is proportional to the square of volatility, which means the penalty does not scale linearly. It accelerates. Double the volatility, quadruple the tax.
Consider what this means in practice. Take a range of portfolios, all earning the same 10% arithmetic average, but experiencing different levels of volatility along the way. The following table shows what happens to $100,000 over 26 years under each volatility regime:
Read the table from top to bottom. Every row has the same arithmetic average: 10% per year. Every row promises the same terminal wealth of $1,191,818. But the geometric reality diverges wildly. At 15% volatility, which is roughly what a broad equity portfolio experiences, the investor arrives at $912,289 instead of $1,191,818. At 25%, the investor arrives at $563,351. Nearly half the promised wealth has been consumed by the volatility tax alone.
Now read the final row. At 45% annualised volatility, a portfolio averaging 10% per year delivers a geometric return of negative 0.12%. The investor loses money. Not because the average return was bad. Not because the strategy was flawed. Because the volatility tax consumed every cent of the arithmetic return and then some. The arithmetic column promises $1,191,818. The geometric column delivers $96,800. The difference is $1,095,018 of wealth that the average promised and compounding refused to deliver.
This is not a theoretical pathology. It is what happens to leveraged products, to concentrated portfolios in volatile sectors, and to any investment whose path to its average is sufficiently turbulent. The average is a mirage shimmering on a straight road. The compound path is the actual road, and it bends.
This relationship has a direct consequence for the leverage decision that most investors miss entirely. Doubling the leverage of any return stream doubles its volatility. But the volatility tax scales with the square of volatility: double the volatility, quadruple the tax. CAGR, which is path-dependent, therefore scales sub-linearly with leverage. Drawdowns, which are linear point observations, scale proportionally. The implication is profound: you cannot manufacture geometric efficiency through leverage. Doubling leverage does not double CAGR. It nearly doubles the drawdown while the CAGR gains diminish with every increment. Episode 10 will quantify this with data from real track records. The ½σ² formula predicts it. The evidence confirms it.
Chart 1: The Volatility Tax: geometric return as a function of annualised volatility for a 10% arithmetic average. The S&P 500 and TTU TF Index are plotted at their actual volatility levels.
Chart 1 makes the same point visually. The horizontal axis is annualised volatility. The vertical axis is the geometric return. The dashed orange line is the arithmetic average: a flat 10% regardless of volatility. The blue curve is the geometric return after the tax. At low volatility, the two are nearly identical. As volatility rises, the gap accelerates. The curve crosses zero somewhere around 45% volatility. Beyond that, more volatility means losing money despite a positive average.
Two points are marked on the curve: the S&P 500 at its actual 15.12% annualised volatility, and the TTU Trend Following Index at its 13.18%. The TF Index sits slightly to the left, in the lower-volatility region of the curve. The difference looks small on this theoretical chart. Over 26 years of real compounding, it is not.
The 26-Year Evidence
The formula is clean. Reality is messy. Let us see what actually happened to money invested at the start of the millennium.
From January 2000 to January 2026, the S&P 500 Total Return Index produced an arithmetic average annual return of 8.89%. If you compounded $100,000 at 8.89% for 26 years, you would expect $916,324. The money actually grew to $747,697. The arithmetic average overstated terminal wealth by $168,627. That is a 23% overstatement: nearly a quarter of the promised wealth simply did not arrive.
The gap of 0.87 percentage points between the arithmetic average (8.89%) and the CAGR (8.02%) looks small in any single year. No annual report would flag it. No quarterly review would mention it. But compounded over 26 years, that seemingly negligible annual leak consumed $168,627 of real wealth from a single $100,000 investment. The S&P 500’s annualised volatility of 15.12% extracted its toll on every dollar, every month, for every one of those 26 years.
The industry showed you 8.89%. Compounding gave you 8.02%. The gap consumed $168,627 of a $100,000 investment. This is not a rounding error. It is the volatility tax, and it never stops collecting.
Now look at the same period through a wider lens. The chart below shows the cumulative growth of $1 invested on 1 January 2000 in four different vehicles: the S&P 500 Total Return Index, Berkshire Hathaway, the TTU Trend Following Index, and a conventional 60/40 portfolio.
Chart 2: Cumulative wealth: $1 invested January 2000 across four benchmarks. Crisis periods (dot-com, GFC, COVID, 2022) shaded in red. Log scale.
The terminal values: $1 in the S&P 500 became $7.48. $1 in Berkshire became $14.11. $1 in the TTU Trend Following Index became $6.67. $1 in the 60/40 portfolio became roughly $4.50.
On terminal wealth alone, the ranking is clear: Berkshire first, S&P 500 second, TF Index third. A careless reading stops here and concludes that trend following underperformed a simple index fund over a quarter-century. That conclusion is wrong, but to see why requires understanding how these numbers are constructed.
The Benchmark Construction Problem
Comparing the TTU Trend Following Index to the S&P 500 is not an apples-to-apples comparison. It is a comparison between two fundamentally different constructions, and the asymmetry between them is significant enough to undermine any naive reading of the terminal values.
The S&P 500 is a trend following index in disguise.
This sounds like a provocation. It is not. Consider what the S&P 500 actually does. It is not a static basket of 500 companies held from inception to eternity. It is reconstituted regularly by a committee at S&P Dow Jones Indices. Companies whose market capitalisation declines fall out of the index. Companies whose capitalisation rises are added. The index is cap-weighted, meaning the largest, most successful companies carry the heaviest weight, and that weight increases automatically as they appreciate. Companies that shrink lose weight. Companies that fail are removed entirely.
Strip away the financial jargon and describe the mechanism plainly: the losers are cut. The winners are retained and their position sizes increase. Failing companies are removed before they can drag the index to zero. Rising companies are added and amplified. This is a momentum and survivorship filter embedded in the index construction itself. The S&P 500 benefits from the same geometric principle that this entire series will explore: it systematically removes the positions that would destroy compounding and concentrates capital in the positions that drive it.
This is, functionally, a form of trend following applied to index membership rather than to position management. The difference is that the S&P 500 version gives you full exposure to every drawdown along the way. When the market falls 50%, as it did during the Global Financial Crisis, the S&P 500 investor absorbs the full loss. The index’s reconstitution process helps with long-term composition but does nothing to protect the compounding base during the crisis itself.
The TTU Trend Following Index, by contrast, is an equal-weighted average of many trend following managers, including weaker performers. It is the equivalent of constructing an equity index by giving every stock equal weight regardless of quality, size, or momentum. If the S&P 500 were reweighted this way, replacing cap-weighting with equal-weighting, its returns would look substantially different (and worse, particularly during periods dominated by mega-cap outperformance). The TF Index is useful as a broad indicator of the trend following industry. It is not the right lens through which to evaluate the best of the process, any more than an equal-weighted stock index represents the best of equity investing.
The more revealing comparison lies in the individual managers who comprise the index. Among the 41 trend following managers in the NilssonHedge database with 20+ year track records, the top quartile, sorted by risk-adjusted return, tells a very different story:
The top quartile of trend followers has a median CAGR of 8.42%, exceeding the S&P 500’s 8.02%. Their median maximum drawdown is 24.3%, less than half the S&P 500’s 50.9%. And their median MAR ratio, which divides CAGR by maximum drawdown and is a crude but powerful measure of geometric efficiency, is 0.334: more than double the S&P 500’s 0.157.
We will return to the MAR ratio in subsequent episodes, and it will become one of the central tools of this series. For now, the essential point is this: the headline comparison between the TF Index (7.55% CAGR) and the S&P 500 (8.02% CAGR) conceals more than it reveals. The indices are not constructed in comparable ways. And when you compare like for like, the best trend following processes match or exceed the S&P 500 on raw return while taking dramatically less drawdown risk. The paths these numbers travelled are structurally different. And in a multiplicative world, the path is not a detail. It is the story.
The Geometric Efficiency Map
If variance is a tax on compounding, then some return profiles pay more tax than others. A portfolio that earns a high arithmetic average but travels a violent path will surrender more of that average to the volatility tax than a portfolio with a comparable arithmetic average and a less volatile path. The ratio of geometric return to arithmetic return is a measure of geometric efficiency: how much of the return the industry reports to you actually survives the journey through compounding.
The chart below plots the arithmetic average annual return against the actual CAGR for each of the 41 trend following managers with 20+ year track records, alongside the S&P 500 and Berkshire Hathaway. The diagonal line represents perfect efficiency: zero gap, zero tax, arithmetic equals geometric. Points sitting close to the diagonal are geometrically efficient. Points sitting well below it are paying heavy volatility tax, losing real wealth that the arithmetic average promised but the compound path consumed.
Chart 3: Arithmetic average return vs CAGR for 41 trend followers, the S&P 500, and Berkshire Hathaway. Points closer to the diagonal retain more return through compounding.
Three observations emerge from this chart.
First, every point sits below the diagonal. This is not an accident. It is mathematically guaranteed. The volatility tax is always positive. Every investment, regardless of strategy, loses some return to the compounding process. The question is not whether you pay the tax. It is how much.
Second, the S&P 500 sits further from the diagonal than many trend followers. The S&P 500’s arithmetic average is 8.89%, and its CAGR is 8.02%, yielding a gap of 0.87 percentage points. The TTU TF Index has a slightly smaller arithmetic average (8.16%) but a smaller gap (0.61 points), producing a CAGR of 7.55%. The TF Index is more geometrically efficient: it retains a larger proportion of its arithmetic return through the compounding process, because its annualised volatility (13.18%) is lower than the S&P 500’s (15.12%).
Third, many individual trend followers cluster closer to the diagonal than the S&P 500. Their return profiles are more geometrically efficient. Less of the raw return is destroyed in the compounding process. This is not a coincidence. It is a consequence of how the trend following process manages risk: cutting losses truncates the magnitude of negative returns, containing drawdowns reduces the variance of the return path, and the resulting return stream retains more of its arithmetic potential as it compounds through time.
This is the first quantitative evidence of a pattern that will define the entire series. Some return profiles are geometrically superior to others. Not because they have higher arithmetic averages. Not because they are more volatile. Because they are structured, by the mechanics of the process that generates them, to lose less of their return to the tax that compounding imposes. The arithmetic average is the marketing number. The geometric average is your money. The gap between them is a function of how the returns arrive, not how large they are.
The Industry’s Quiet Fraud
The investment industry knows every word of this. It quotes arithmetic averages anyway.
Fund fact sheets report arithmetic average returns. Pension projections compound arithmetic assumptions forward. Financial planning software uses arithmetic averages as inputs to compound growth calculations. In every case, the projected terminal wealth is higher than what geometric compounding will actually deliver. The gap is not a rounding issue or a simplification. It is a structural feature of how the industry communicates returns, and it systematically flatters performance.
Consider the implications for a pension fund. A defined-benefit plan that assumes an 8% arithmetic return on its equity allocation will project terminal assets that exceed what 8% CAGR actually delivers, by an amount that depends on the volatility of the equity portfolio. For a typical equity allocation with 15% annualised volatility, the volatility tax consumes roughly 1.1 percentage points per year. Over a 30-year accumulation phase, that seemingly modest overstatement compounds into a meaningful funding shortfall: the pension promises benefits calculated on an arithmetic projection while the portfolio earns geometric reality. This is one structural reason why pension deficits persist across the developed world. The mathematics are not wrong. The projections are using the wrong average.
But the overstatement of arithmetic averages is not an isolated error. It is a symptom of a deeper misdirection. The entire performance narrative of the investment industry is constructed around selection: which stocks were picked, which sectors were favoured, which calls proved right. Fund managers are hired for their picks. They are fired for their picks. Their investor letters celebrate their picks. Magazine covers feature their picks. Yet the volatility tax demonstrates something quietly devastating to this narrative: terminal wealth is governed less by what was selected than by the geometric properties of the path those selections produced. Two managers can hold identical stocks and arrive at radically different terminal wealth if their position sizing, their drawdown management, and their volatility profiles differ. The industry celebrates selection because selection makes for compelling stories. Geometry does not. But geometry is where the money lives, and the arithmetic average is the first of many tools the industry uses to keep your attention on the story instead of the structure.
The most extreme case study is the Nikkei 225. The Japanese equity market peaked at 38,957 in December 1989. It did not recover that level until February 2024: thirty-four years later. During those decades, the arithmetic average annual return was positive across many rolling windows. An investor reviewing annual performance summaries would have encountered encouraging numbers. But the geometric reality was a catastrophe: a market that fell roughly 80% from its peak and spent more than three decades recovering ground that the volatility tax, the asymmetry of drawdowns, and the mechanics of compounding conspired to prevent it from reclaiming. The arithmetic average told a story of gradual recovery. The compound path told a story of a lost generation of Japanese wealth.
The Nikkei is an outlier in its severity but not in its mechanism. Every equity market in every era exhibits the same structural gap between arithmetic and geometric returns. The magnitude varies. The direction never does. The arithmetic average always overstates. The compounding process always collects.
The Running Ledger
Throughout this series, we will track a single thought experiment: $100,000 invested on 1 January 2000, allocated to four paths. The ledger updates at the close of each episode. Here is where we stand at the end of Episode 1:
The fourth row is the ghost in the machine: the $916,324 that the S&P 500’s arithmetic average promised but that geometric compounding refused to deliver. That phantom of $168,627 is the cumulative cost of the volatility tax applied to a single $100,000 investment over 26 years. It is wealth that exists in projections, in fund marketing, in pension assumptions, and nowhere else.
Notice one number in the rightmost column. The TTU Trend Following Index pays the smallest volatility tax of the three real investments: 0.61% per year versus 0.87% for the S&P 500 and 1.09% for Berkshire Hathaway. Despite earning a lower arithmetic average, the TF Index retains a higher proportion of that average through compounding. It is, on this measure, the most geometrically efficient of the three. Why this is so, and what it implies for the construction of long-term wealth, is the subject of the episodes that follow.
The Bridge
The volatility tax explains the gap between the return you are quoted and the return you earn. But it raises a deeper question, one that cuts to the foundations of how wealth is created and destroyed.
If variance is a cost, and the cost accelerates with the square of volatility, then losses are not the mirror image of gains. A 50% loss is not offset by a 50% gain. A 50% loss requires a 100% gain to recover. A 75% loss requires a 300% gain. The damage is not symmetric. It is not linear. It is exponential, and the asymmetry compounds over time with every drawdown the portfolio absorbs.
This asymmetry has a name. It is called convexity, and it is the reason why, in a multiplicative world, protecting the compounding base from deep losses matters more, mathematically, than capturing the largest gains. It is the reason why an investor who avoids the worst outcomes will, over sufficient time, compound more wealth than an investor who captures the best ones.
This series is structured in four acts. Act I (Episodes 1 to 4) dismantles the mathematical framework on which conventional investing rests: the arithmetic average, the treatment of drawdowns, the Sharpe ratio, and the Gaussian model of returns. Act II (Episodes 5 to 8) identifies the mechanism: why markets produce trends, how the systematic process harvests them, and why its strongest returns arrive during the moments of maximum danger for every other strategy. Act III (Episodes 9 to 12) deploys the evidence across 41 independent track records spanning 26 years, addresses the survivorship and start-date objections, and quantifies the costs without flinching. Act IV (Episodes 13 to 15) converts the framework into practice: how to evaluate managers, how to construct the portfolio, and what the full body of evidence reveals about the nature of wealth creation itself. Each episode builds on the one before it. The argument is cumulative. By Episode 15, the six propositions and two corollaries that govern geometric wealth creation will have been established not as theory but as demonstrated fact.
In Episode 2, we will map this asymmetry in full: the non-linear cost of drawdowns, the exponential recovery problem, and the first evidence from 26 years of real data that a particular investment process is built, from the ground up, to exploit it.
Data and Sources
All performance data from the NilssonHedge Trend Following Performance Database (January 2000 to January 2026). Benchmarks: S&P 500 Total Return Index, Berkshire Hathaway (BRK-A equivalent), TTU Trend Following Index (equal-weighted composite of reporting trend following managers), Vanguard Balanced Index (60% equity / 40% bond proxy). Individual fund returns are reported net of management and performance fees. The NilssonHedge dataset contains 51 trend following funds with return data in this period, of which 41 have continuous track records of 20 years or more. The volatility tax approximation (½σ²) is a second-order Taylor expansion of the logarithmic growth rate; it is exact for lognormally distributed returns and a close approximation for real-world return distributions. Nikkei 225 reference: peaked at 38,957.44 on 29 December 1989; first closed above that level on 22 February 2024 at 39,098.68.
Want the theoretical foundation for why trend following works?
The Fractals of Finance: Determinism, Adaptation and the Geometry of Markets
The book explores the full architecture of feedback, fat tails, and fractal structure in financial markets, and what it means for how we trade, invest, and understand risk.
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Want a practical field manual for trading trends and capturing outliers?
The Aussie Turtles Trend Following Guide: A Field Manual for Hunting Outliers adapts the timeless principles of the original Turtle traders into a systematic, rules-based approach for modern markets. Co-authored with Adam Havryliv.
Available now on Amazon in paperback, hardcover, and Kindle.