An investment’s average result ALWAYS exceeds – often greatly – its most likely result. I explain how and why, and what you can do about it.

**Most Investors and Traders-Speculators Underperform**

It’s long been widely known: the long-term returns of most managed investments fail to match – never mind exceed – benchmarks such as the S&P 500 and S&P/ASX 200 indexes. S&P Global’s *SPIVA Institutional Scorecard Year-End 2023*, released on 29 July, provides the most recent figures: “more than 70% of funds underperformed their respective benchmarks over the 10-year period ending 31 December 2023. After deducting fees, underperformance rates increased above 80%” (for Leithner & Company’s results since 1999, see our web site).

In contrast, and not least because publicly-available data are relatively scant, few people realise that an even greater percentage of individual (“DIY”) investors underperform. Brad Barber and Terrance Odean, “Trading Is Hazardous to Your Wealth: The Common Stock Investment Performance of Individual Investors” (*Journal of Finance*, April 2000, pp. 773-805), remains the most rigorous and readable study.

Moreover, there’s plenty of compelling logic (to which I’ll add) that the vast majority of traders-speculators don’t merely miss key benchmarks: they destroy considerable percentages of their capital. These supposedly smart people, it seems, repeatedly do stupid things (see also Does high IQ make a better investor? 20 November 2020).

For these reasons, advocates of index funds urge that many DIY and actively-managed fund investors, as well as most traders-speculators, embrace low-fee funds that replicate major indexes. Warren Buffett agrees: although “diversification is for the know-nothing investor … by periodically investing in (a low-fee) index fund … the know-nothing investor can actually outperform most investment professionals.”

Providers of index funds contend that switchers’ returns, net of costs and fees, will match a relevant benchmark. That’s true if you’re a long-term, “buy and hold” investor. But many people aren’t: according to the ASX’s *Australian Investor Study 2023*, owners of SMSFs trade an average of 1.2 times per month and HNWIs transact 3.9 times per month.

On that basis, it’s reasonable to infer that many “investors” are actually traders and speculators. If so, then index funds’ core claim is occasionally true but is typically false (see also Index funds’ key flaws – and how we overcome them, 23 October 2023).

Furthermore, thematic exchange-traded funds usually underperform – and sometimes crash. All of the major artificial-intelligence-themed ETFs, to cite a topical example, have lost money this year. This “AI fund disaster,” writes James Mackintosh, “should be a cautionary tale for buyers of thematic ETFs.”

“Put simply,” he concludes, “you probably won’t get what you want, you’ll likely buy at the wrong time and it will be hard to hold for the long term … Even investors who really want to commit … for the long run often find it hard, as so many funds are wound up, merged or change strategy when they go out of fashion” (see “How to Lose Money on the World’s Most Popular Investment Theme,” *The Wall Street Journal*, 1 September). “There’s an ETF for nearly everything,” adds Jason Zweig (“What’s Left to Be ETF’d?” 13 September), “but good ones are rare” (see also “Investment Industry Loves Active ETFs. You Probably Shouldn’t.” *The Wall Street J*ournal, 7 October).

Morningstar’s conclusion thus applies to index funds, active and thematic ETFs as well as actively-managed funds: “when a fund achieves a 7% return over a set period that usually doesn’t mean that its investors became 7% richer during the same period.”

Its *Mind the Gap* study found that fund investors’ return over the 10 years to 31 December 2020 averaged 6.3% per year. That’s “about 1.7 percentage points (per year) less than the total returns their investments generated over that span. This shortfall, or ‘gap,’ stems from inopportunely timed purchases and sales of fund shares …” (see “Why the Fund You Own Got Better Returns Than You,” 16 November 2023).

Rather than buy and hold, fund “investors” tend to buy high and sell low (for details, see Amy Arnott, CFA, “Why Fund Returns Are Lower Than You Might Think,” 30 August 2021). That’s hardly a recipe for success!

Morningstar’s results corroborate others’ findings over the decades. Most notably, Jason Zweig and a team of academic researchers analysed data from more than 6,900 U.S.-based funds during 1998, 1999, 2000 and 2001. By taking into consideration the money which investors added and withdrew from funds, they calculated investors’ (as opposed to funds’) return.

“To a remarkable degree,” they found, “investors underperformed their funds’ reported returns – sometimes by as much as 75 percentage points a year. The sheer magnitude of the difference we discovered between the total returns earned by funds and the results captured by the average shareholder is shocking and tragic” (see “What Fund Investors Really Need to Know,” jasonzweig.com, 25 November 2016).

Zweig recently reviewed evidence from more recent analyses. He concluded: “investors typically underperform their investments, not just in mutual funds and ETFs, but in hedge funds and stocks as well” (for details, see “Messing Up the Closest Thing to a Sure Thing in the Stock Market,” *The Wall Street Journal*, 23 August 2024).

Most funds and DIY investors, in short, underperform major benchmarks (speculators grossly so) – and most fund investors underperform their funds.

**Why Underperformance Is Rife**

Apart from costs and fees, the biggest reason by far is that traders-speculators, DIY investors and funds managers are human. As such, they’re typically overconfident – sometimes grossly so – and thus, often unwittingly, take excessive risks (see also Why you’re probably overconfident – and what you can do about it, 14 February 2022).

In *The Wealth of Nations* (1776), Adam Smith detailed “the overweening conceit which the greater part of men have of their own abilities.” Much more recently, Richard Thaler told *Barron’s* (4 December 2020): “the biggest mistake people make is overconfidence.” Indeed, he and Werner De Bondt reckon that “perhaps the most robust finding in the psychology of judgement is that people are overconfident.”

Arrogance eventually crimps – indeed, crushes – returns. Occasionally, however, most people and funds get lucky: they generate short-term results which greatly exceed their benchmarks and peers. “For a large majority of fund managers,” concluded Daniel Kahneman (*Thinking Fast and Slow*, Farrar, Straus & Giroux, 2013), “the selection of stocks is more like rolling dice than playing poker … The successful funds in any given year are mostly lucky; they have a good roll of the dice.”

Lucky funds attract considerable attention and unwarranted praise from gullible journalists – and often substantial inflows from investors who apparently believe that exceptional results will continue. By their very nature, however, they’re fleeting.

As a result, lucky funds’ returns soon slump – and unduly risky funds eventually crash. Investors who’d been exuberant become despondent, and more than a few of them sell. Incurious journalists take no notice; instead, they naively and irresponsibly laud the next outperformer, and the cycle repeats.

S&P Global’s *U.S. Persistence Scorecard Year-End 2023*, published on 14 May, substantiates this pattern: “consistent outperformance is typically fleeting. Among top-quartile funds within all reported active domestic equity categories as of December 2019, not a single fund remained in the top quartile over the next four years.”

This isn’t merely an American phenomenon; it applies here, too. According to S&P Global’s *Australia Persistence Scorecard: Year-End 2023*, released on 6 May,** **“almost no Australian funds … remained in the top performance quartile within their category over five consecutive years ending December 2023. The results did not improve significantly by lowering the bar from the top quartile to the top half: the proportion of funds remaining in the top half over five consecutive years was smaller than would be expected if the performance were completely random.”

There’s another crucial – and almost totally unremarked – reason why many individual investors generally underperform, speculators generate huge losses and managed funds and DIY investors fail to match their benchmarks. Probably unwittingly but nonetheless systematically, they overstate their returns. This exaggeration exacerbates their overconfidence; and hubris reinforces the poor behaviour which begets mediocre – and worse – results.

Why are DIY investors’ returns usually lower (and traders-speculators’ far lower) than they believe? Why are the returns of managed funds almost always lower than they assert? How do they typically overstate their results? John Allen Paulos, professor of mathematics at Temple University and the author of *A Mathematician Plays the Stock Market* (Basic Books, 2003, p. 96), “illustrates the crucial difference between the arithmetic mean and the geometric mean of a set of returns.”

According to Paulos (p. 99), the ubiquitous use of arithmetic means to quantify returns, and the virtual banishment of more appropriate geometric mean, “explain why a majority of investors receive worse-than-average returns and why some mutual fund companies misleadingly stress their average returns … The reason is that the average or arithmetic mean of different rates of return is always greater than the geometric mean of these rates of return, which is also the median rate of return.”

In this article, I unpack and elaborate Paulos’ crucial insight. I also analyse a prominent Australian equity fund’s results since 2017. Expressed as geometric rather than arithmetic means, its results aren’t merely much lower than it maintains: at best they’re mediocre and at worst they’re abysmal.

**Some Crucial Disclaimers**

Any managed fund which for whatever reason overstates its results is fair game for dispassionate criticism. As we’ll shortly see, the arithmetic mean (“average”) of a series of returns always exceeds – often significantly and sometimes greatly – its geometric mean (compound annual growth rate (CAGR)). Yet the arithmetic mean is the standard measure of returns, and the geometric mean is virtually unknown. Consequently, virtually all managed funds – and DIY investors – exaggerate their results.

I assume that this is mostly inadvertent; that is, they’re blindly following the custom of using arithmetic means to quantify their returns.

In one sense this is understandable. Averages are much simpler to compute than CAGRs; moreover, the typical person readily understands the concept and calculation of the average – and probably has never heard of, let alone is able to calculate, a geometric mean.

Yet I strongly doubt that the pervasive use of arithmetic (and banishment of geometric) means to quantify returns is entirely innocent: it’s essential to bear in mind that the average of a series of returns always exceeds its compound rate of growth.

Finally, although I criticise the ubiquitous use of arithmetic means to quantify returns, I’ve nothing in particular against the fund which I’ll analyse (it’s hardly the only one whose results, stated properly, are mediocre). That’s why it’ll remain anonymous and I’ll refer to it as “Fund X.”

**Start with $10,000 and End with $1.4 million – or $1.95?**

How and why does an investment’s *average return* always exceed its *most likely return* – often significantly and sometimes enormously? Assume that you’re a trader-speculator; specifically, you purchase and quickly sell stocks whose prices are extremely volatile. As I’ve repeatedly demonstrated, during periods of at least 12 and up to 18 months it’s effectively impossible to foresee by how much and in which direction stocks’ prices (and indexes’ levels) fluctuate; but for the sake of simplicity let’s consider three scenarios. During any given week, and extending Paulos’ example (pp. 95-99),

- there’s a 50% chance that these very volatile stocks’ prices zoom 80%, and a 50% chance that they plunge 60%;
- the likelihood is 50% that their prices rise 60%, and 50% that they fall 40%;
- there’s a 50% chance that their prices increase 40%, and a 50% chance that they decrease 20%.

Each week for a year, each Monday morning you buy shares of one of these highly volatile stocks – and you sell them just before the market closes on Friday afternoon. Consequently, during each week in Scenario #1, there a 50-50 chance you’ll enjoy a whopping gain of 80% – and that you’ll suffer a crushing loss of 60%. Your initial investment is $10,000, and each week you reinvest the cumulative proceeds from preceding weeks.

Given these assumptions, what’s your average rate of return? The answer is simple, and surely everybody knows it: (80% – 60%) ÷ 2 = 10% per week. Everybody also knows that Scenario #2 and Scenario #3 also generate average returns of 10% per week.

This average return generates a startling end result. If you begin with $10,000 then (ignoring brokerage and other costs, tax, the liquidity of the shares you buy and sell, etc.) at the end of the 52^{nd} week under Scenario #1 your capital will skyrocket to an average of $1,420,329.32.

It bears repeating: this is the *average* result. Let’s now ask a crucial question that nobody ever seems to ask: what’s the *most likely* result? The answer is even more astonishing: in Scenario #1, you’d lose more than 99.9% of the initial investment; after 52 weeks, your $10,000 would shrink to just $1.95!

This most likely result is also the median result: under these assumptions, half of the time your end result would be more than $1.95, and the other half it’d be less. On average, you’ll turn each $1 into $142 and make a small fortune; it’s most likely, however, that you’ll lose virtually everything.

Under Scenario #2, after 52 weeks it’s most likely that your initial investment of $10,000 would shrivel to $3,459.81; under Scenario #3, however, it’d rocket to $190,400.72. These results emerge when your speculations shrink (or grow) at a rate equal to the geometric mean over 52 weeks of (1) 80% and -60%, (2) 60% and -40% and (3) 40% and -20%.

What on earth is happening here? Why the colossal disparity in Scenario #1 between the average (arithmetic mean) result, an astounding gain of more than $1.4 million, and the most likely (and, as we’ll see, geometric mean) result – a near-total loss of the initial $10,000, leaving a paltry $1.95? Why the huge differences among the three scenarios’ most likely results?

I’ll unpack the mathematics shortly; in the meantime, it’s clear that in Scenario #1 the typical speculator will gain 80% during approximately 26 (i.e., 50% of 52) weeks and lose 60% during approximately 26 weeks. The lucky speculator, in sharp contrast, will receive a return of 80% during considerably more than 26 weeks – and thus lose 60% during considerably fewer than 26 weeks. Her astronomical return boosts the average. The unlucky speculator, on the other hand, will lose 60% during significantly more than 26 weeks – but his losses cannot exceed his original $10,000.

“In other words,” concludes Paulos (p. 97), “the enormous returns associated with disproportionately many weeks of 80% growth skew the average way up, while even many weeks of 60% shrinkage can’t drive an investment’s value below $0. In this scenario … the average worth of your $10,000 after one year is $1.4 million, but it’s most likely worth is $1.95. Which results are the media likely to focus on?” Which, I add, will transfix traders-speculators?

**A Brief Aside for Two Definitions**

“The arithmetic mean of N different rates of return,” writes Paulos (p. 96), “is what we normally think of as their average; that is, their sum divided by N. The geometric mean of N different rates of return is equal to that rate of return that, if received N times in succession, would be equivalent to receiving the N different rates of return in succession.” The geometric mean return is what’s commonly known as the compound rate of return.

I’ll spare you the details, but a mathematician could easily derive the formula for the geometric mean from the formula for compound interest. This mean, Paulos adds, “is equal to the N^{th} root of the product ((1 + first return) × (1 + second return) × … × (1 + N^{th} return)) -1.”

Got that? For investors, it’s not the mathematical details but the general concept and its consequences which are crucial.

**Returning to Our Example …**

Why the colossal disparity in Scenario #1 between the average and arithmetic mean result (an astounding $1.42 million) and the most likely and geometric mean result (a paltry $1.95)? Following Paulos, let’s examine what happens to an initial investment of $10,000 during its first two weeks. There are four equally-likely possibilities: (1) the investment zooms 80% during both weeks; (2) it rockets 80% during the first and collapses 60% during the second; (3) it decreases during the first and increases during the second; or (4) plummets during both weeks.

The capital of lucky speculators (category 1) will zoom by a factor of (1 + 0.8) × (1 + 0.8) = 3.24; after two weeks, their initial $10,000 therefore grows to $10,000 × 3.24 = $32,400. The capital of those in category 2, however, shrinks by (1 + 0.8) × (1 – 0.6) = 0.72; after two weeks, their initial $10,000 becomes $10,000 × 0.72 = $7,200. So does category 3 (the order of multiplication, but not its product, changes). Finally, the stake of unlucky speculators (category 4) collapses; after two weeks, their initial $10,000 shrivels to $10,000 × 0.4 × 0.4 = $1,600.

Consider these four possibilities: ($32,400 + $7,200 + $7,200 + $1,600) ÷ 4 = $12,100. That’s the speculations’ average market value after two weeks. It’s equivalent to a compound rate of return of 10% per week, since $10,000 × 1.1 × 1.1 = $12,100. After 52 weeks, the investment’s average value is thus $10,000 × (1.1)^{52} = $1,420,429.32.

Notice that after two weeks the average result ($12,100) is a gain of 10% per week; but the most frequent – and thus most likely – result ($7,200) is equivalent to a loss of ca. 15% per week, i.e., $10,000 × (1 – 0.15)^{2} ≈ $7,200.

This is why traders-speculators of volatile assets, such as small cap and tech stocks, crypto-currencies, etc., typically lose heavily: they obsess about averages (which seemingly ensure large gains) and ignore probabilities (which often produce hefty losses); hence the longer they speculate the more they lose (see also Philip Newall and Leonardo Weiss-Cohen, “The Gamblification of Investing: How a New Generation of Investors Is Being Born to Lose,” *Int J Environ Res Public Health*, vol. 19, no. 9, May 2022).

Given our assumption that there’s a 50% chance that during any given week volatile stocks rise 80%, the most likely result is that the speculator’s pick rises during 52 × 0.5 = 26 weeks; given that there’s a 50% chance that during a given week such stocks fall 60%, it’s also most likely that his selection declines during 26 weeks.

Accordingly, the investment’s most likely value after 52 weeks is $10,000 × 1.8^{26} × 0.4^{26} = $1.95. In other words, the geometric mean of 80% and -60% is √(1.8 × 0.4) – 1, which equals approximately -0.15. Every week, the most likely result is that this speculator’s investment loses ca. 15% of its value, just as it did in the first two weeks; hence after one year, $10,000 × (1 – 0.15)^{52} ≈ $1.95.

By varying our assumptions we obtain very different results. Under Scenario #2, after 52 weeks the initial investment of $10,000 shrinks to $3,459.81; under Scenario #3, however, it zooms to $190,400.72. The geometric mean of 60% and -40% equals approximately -0.02. Every week, on average, this speculator’s initial grubstake of $10,000 loses ca. 2% of its value, and $10,000 × (1 – 0.02)^{52} ≈ $3,459.81. And the geometric mean of 40% and -20% ≈ 0.0583. Each week, on average, the value of this speculator’s stake rises almost 6%; after 52 weeks we thus have $10,000 × (1.0583)^{52} ≈ $190,400.72.

Despite these very different results in the three scenarios, says Paulos (p. 99), “the principle holds true: the arithmetic mean of the returns far outstrips the geometric mean of the returns, which is also the median (middle) return as well as the most common (modal and hence the most likely) return.”

Of course, no stock or collection of stocks ever rises by a fixed percentage with a constant probability x, and falls by a fixed percentage with an invariant probability y, over some period z. Nonetheless, concludes Paulos, our results thus far “have much more general importance than it might appear.” Specifically,

“They explain why a majority of investors receive worse-than-average returns and why some mutual fund companies misleadingly stress their average returns. Once again, the reason is that the average or arithmetic mean of different rates of return is always greater than the geometric mean of these rates of return, which is also the median rate of return” (see also pp. 132-134).

**The More Volatile the Returns, the More Misleading the Arithmetic Mean Becomes**

To appreciate the full significance of Paulos’ insight, let’s analyse an actual example. But before we do so, recall the results across the three scenarios in our hypothetical example: an initial investment of $10,000 leads to a near-total loss in Scenario #1, a loss of more than 65% in Scenario #2, but a huge (19-fold or 1,800%) gain in Scenario #3.

There’s a crucial pattern here: holding constant the arithmetic mean in our three scenarios (10% per week), the greater is the volatility of returns over time (Scenario #1 is more volatile than Scenario #2, and Scenario #2 is more volatile than Scenario #3), the greater is the disparity between the arithmetic and geometric mean, and thus between the average and most likely return.

In other words, the more volatile over time are returns (in particular and as we’ll see, the greater is the number and magnitude of negative returns in a series of returns) the lower is the geometric mean (i.e., compound rate of) return – and therefore the stronger is the incentive for managed funds to stress arithmetic mean returns and ignore geometric mean returns.

**A Real-Life Example**

Figure 1 compares the growth since January 2017 – assuming the reinvestment of distributions and dividends, and adjusted for issues and buybacks of shares – of investments of $1,000 in the All Ordinaries Index (or an index fund that mimics it) and Fund X (recall the Disclaimer). $1,000 invested in the Index in January 2017 grew to $1,828 in June 2024. That’s a compound annual growth rate (CAGR) of 8.4% per year. $1,000 invested in the Fund grew to $1,390 in June 2024. That’s a CAGR of 4.5% per year.

**Figure 1: ****Investments of $1,000 in the All Ordinaries Index and Fund X, ****Distributions Reinvested, January 2017-June 2024**

As a brief but important aside, Leithner & Company and a few other investors present their results as geometric means (commonly called CAGRs) in a form similar to Figure 1. But most don’t, and we can guess why: the results wouldn’t flatter them!

The Index’s geometric mean rate of return is 8.4% per year over 7.5 years; that’s because $1,000 × (1.084)^{7.5} ≈ $1,828. In contrast, the Fund’s geometric mean return is just 4.5%: $1,000 × (1.045)^{7.5} ≈ $1,390.

Over the 7.5 years since January 2017, Fund X, like most managed funds, has failed to match the Index. In particular, since mid-2018 it’s lagged badly and continuously. I haven’t read all of X’s commentaries, reports, etc., during these years, but I’ve skimmed a reasonable sample – and nowhere does it confess this crucial failure’s total magnitude. Instead, it lauds arithmetic means of its returns since mid-2020 – which (what a surprise!) are much higher than its geometric mean return since 2017.

Particularly noteworthy is Fund X’s gross underperformance during the two years to March 2020: the investment collapsed from $1,158 in March 2018 to just $458 in March 2020: that’s a plunge of 60% and a CAGR of -37.1% per year over the two years. In sharp contrast, during this interval the Index fell from $769 to $727: that’s a decrease of just 5.5% and a CAGR of just -2.8% per year for two years.

Although not in terms and numbers like those in the preceding paragraph, Fund X has discussed its underperformance during the COVID-19 crisis. But it’s regarded this gross underperformance as a “one-off” – as merely a very unlucky event – rather than as an egregious example of comprehensive underperformance.

The truth is that Fund X has underperformed the Index since 2017 because it’s consistently – that is, over moving 12-month, 24-month, 36-month, 48-month and 60-month periods – underperformed (Figure 2).

**Figure 2: Rolling Total CAGRs, ****All Ordinaries Index and Fund X, January 2017-June 2024**

Most managed funds present their results in a format which superficially resembles Figure 2. Crucially, however, they almost always present them as arithmetic rather than geometric means. They thereby flatter themselves – and withhold crucial information from investors.

In its commentaries, reports, etc., Fund X has conspicuously avoided any mention – never mind confession and explanation! – of its pervasive underperformance.

The Fund’s and the Index’s one-month returns are indistinguishable; the other returns, however, heavily favour the Index. Moreover, the Index’s returns over 12-month to 60-month intervals are remarkably stable, whereas the Fund’s are erratic; in particular, its five-year returns are – next only to its one-month returns – its lowest.

Why has Fund X comprehensively underperformed? Here’s another key point that its commentaries, reports, etc., diligently ignore: the volatility of its returns greatly exceeds the Index’s. Figure 3 plots what Fund X omits even to mention and what it and the vast majority of managed funds fail to report: the variability of their rolling one-month, 12-month, … and 60-month returns. In most cases, and for all intervals of 12 months or more, the standard deviations of X’s returns are more than twice as great as the Index’s.

**Figure 3: Standard Deviations of Rolling Total CAGRs, ****All Ordinaries Index and Fund X, January 2017-June 2024**

Why has Fund X comprehensively underperformed? It’s not just that the overall volatility of its results is significantly greater than the Index’s: the frequency and magnitude of its downward negative results greatly exceeds the Index’s.

I computed the frequency and the magnitude of Fund X’s and the Index’s negative results during six rolling intervals since January 2017. Table 1 summarises the results. Fund X generates negative results much more frequently than the Index; and X’s results during negative months are much worse than the Index’s.

**Table 1: Frequency and Magnitude of Negative Results, ****Six Intervals, January 2017-June 2024**

Notice that as the interval of time lengthens the Index’s frequency of negative months declines rapidly: although it falls in almost one-third of the one-month intervals, its CAGR in just 3% of rolling 24-month intervals is negative. In sharp contrast, from periods of one to 24 months Fund X’s percentage of negative months slightly increases; even worse, from periods of 36 to 60 month it almost doubles. Astoundingly, over most intervals, Fund X generates negative results almost half of the time!

Fund X, in short, doesn’t merely generate losses much more frequently than the Index, and its losses aren’t merely significantly greater than the Index’s: its negative returns persist far longer than the Index’s.

Finally, Figure 4 plots the Fund X’s and the Index’s worst one-month, 12-month, … and 60-month CAGRs since January 2017. Over all of these intervals, the Fund’s worst CAGR is dramatically worse than the Index’s. Between February and March 2020 – in other words, at the nadir of the COVID-19 panic, the Fund plunged an astounding 41%. The Index also fell severely but less than half as much (-21%).

**Figure 4: Worst Return, Rolling Compound Rates of Total Return, ****All Ordinaries Index and Fund X, January 2017-June 2024**

Similarly, during the 12 months to March 2020 the Fund crashed 54%, whereas the Index sagged just 15%. A positive rate of return compounds capital, and a negative one shrinks it. Notice that all of the Fund’s worst CAGRs are negative; note in particular that its worst five-year CAGR is -2.7%. In sharp contrast, the Index’s worst five-year CAGR is +6.4%.

S&P Global’s *Australia Persistence Scorecard Year-End 2023* asked: “can investment results be attributed to skill or luck? Genuine skill is more likely to persist, while luck is random and fleeting. Thus, one measure of skill is the consistency of a fund’s relative performance.”

On that basis, it’s clear that Fund X’s results since January 2017, both in absolute terms and relative to the Index, stem from occasional good luck and constant lack of skill.

**What Are Investors in Managed Funds To Do?**

An investment’s average and its most likely return ALWAYS differ – often significantly and sometimes dramatically. In particular, the arithmetic mean (“average”) of a series invariably exceeds its geometric mean (CAGR). Unsurprisingly, the arithmetic mean is the standard measure of returns and the geometric mean is virtually unknown.

Consequently, virtually all managed funds and DIY investors – I assume unwittingly but not completely innocently – overstate their returns. That overstatement, I suspect, further inflates their overconfidence.

In response, what can investors do? Firstly, those who invest via managed funds (including ETFs) might request from their funds information like that which I’ve presented above. The chance that they receive it is probably miniscule: most funds probably don’t possess such information, and even if they did I strongly doubt they’d disclose it. Why confess, in effect, that you’ve been exaggerating your returns?

What might investors do? Secondly, investors can easily obtain the data they require and analyse it themselves. I downloaded Fund X’s monthly prices and half-yearly distributions; anybody can do it just as easily for whatever fund he wishes to analyse.

Clearly, however, few investors have the background, time and energy to collate and analyse such data. That leaves a third option: seek the small number investment vehicles which present and analyse their results in the form of geometric rather than arithmetic means. I can think of one!

**What To Do? A Hypothetical but Realistic Example**

Consider a couple, family trust, SMSF, etc., which possesses a long-term perspective and the brains – but not the desire, time or energy – to manage their own capital. What, in light of Paulos’ insights and my analysis of Fund X’s returns, should they do?

First, ignore the latest performance ranking – or else use it for one purpose only: to exclude recent “outperformers” from consideration.

Mark Hulbert has analysed decades of data (“The Year’s Fund Returns Are In – Do They Matter?” *The Wall Street Journal,* 7 January 2018) and concludes that managers who possess strong records of “long-term performance … are hardly ever at the top or bottom of the calendar-year rankings. Slow and steady really does win the race.”

Hulbert’s point is fundamental – yet journalists (who every six months mindlessly cheer short-term outperformers) resolutely ignore or deny it.

To grasp its significance, consider as a hypothetical yet realistic example the three managed funds in Table 2. (As an aside, the order of annual results doesn’t affect their three-year returns.) Fund A greatly outperforms in Year 1 – at whose end, and perhaps as a result of laudatory media reports, it likely attracts hefty inflows.

**Table 2: ****Three Funds over Three Years: the Steady Tortoise Beats the Erratic Hares**

Alas, Fund A incurs a hefty loss and wins the wooden spoon in Year 2 – which prompts strong outflows. Speculators who thought they were investors (or investors advised by speculators) bought its units high and sold them low – which is hardly a recipe for success! Fund B greatly lags in Year 1 but excels in Years 2 and 3.

In contrast, in no single year does Fund C lead the field; indeed, each year its return is just one-half of the top-ranked fund’s. Furthermore, in Year 3 it trails the others. Yet two key points distinguish it: first, it never incurs a loss; second, from year to year its results are the steadiest (i.e., its standard deviation is the lowest by far).

Over the three-year period, these traits make all the difference: Fund C’s three-year return, expressed as a geometric mean (CAGR), handily exceeds A’s and B’s.

Ignoring costs, fees, taxes, etc., $100 invested in Fund C at the beginning of Year 1 compounds to $100 × 1.075 × 1.075 × 1.0625 = $122.79 at the end of Year 3. That amount exceeds Fund B’s cumulative total ($116.44) by more than 5% and Fund A’s ($115.50) by more than 6%.

If this disparity persists, as time passes Fund C will leave A and B ever further in its wake.

What to do? Secondly, advises Hulbert, focus on those managers, strategies and vehicles with excellent long term results – which, crucially, are expressed as CAGRs rather than simple averages. That certainly eliminates Fund X! “The clear implication,” he elaborates, is that “you improve your chances of picking a (winner) by focusing on performance over periods far longer than one year.” How much longer? Hulbert continues:

“Our analysis … suggests that even 10 years isn’t enough. Only when performance was measured over at least 15 years were there better-than-50% odds that a top performer would be able to repeat. (Moreover), a top performer over the previous 15 years (is) unlikely to be at the top of the rankings in any given calendar year …”

To Hulbert’s point I’d add an elaboration. Locate investment managers with a track record of 20 or more years, and then ascertain:

- did their results during the Dot Com Bubble and the boom years before the GFC suggest that they took undue risks?
- what about the Dot Com Bust and the GFC? The COVID-19 panic? Did they repent during the bust for their sins during the boom?
- how many years did they require to recoup these losses?

Above all, locate managers with the fewest months, years, etc., of negative returns, and those whose negative returns are smallest. In other words, find those who mind the downside; the upside will mind itself.

The problem, of course, is that few managers possess continuous track records of 20 or more years. Moreover, many grasp for the upside, discount and even ignore the downside – and eventually suffer the consequences.

The good news is that your list of candidates certainly won’t be long; hence your choice probably won’t be difficult!

**Implications for Conservative-Contrarian Investors**

Charlie Munger, who until his death in November of last year had for more than six decades been Warren Buffett’s closest confidant and since 1978 Berkshire Hathaway’s vice-chairman, repeatedly acknowledged his intellectual debt to the Prussian mathematician Carl Jacobi. The latter famously advised: *“man muβ immer umkehren”* (loosely translated: “invert, always invert”). Jacobi, said Munger, “knew that it is in the nature of things that many hard problems are best solved when they are addressed backwards.”

Munger meant that you shouldn’t consider problems exclusively or even primarily from one conventional point of view: instead, you should approach them counter-intuitively (“backward”) as well as conventionally (“forward”). Inversion enables you to clarify and uncover novel solutions. “Indeed,” reckoned Munger, “many problems can’t be solved forward.”

If, for example, you want to promote innovation, the conventional approach is to think “forward” – that is, devise and implement measures that allegedly foster innovation. If, however, you invert the problem – that is, think “backward” – you identify and eliminate things that you and others are presently doing that discourage innovation.

Fostering innovation isn’t a simply matter of adopting new and good practices; it’s also about eradicating old and bad ones that stifle innovation. The latter might be easier and more effective than the former; indeed, until you achieve the latter you probably can’t achieve the former.

Thinking backwards, it’s clear that the distinctive feature of a sensible investor and fund, and the key to solid long-term returns, is NOT the impulsive, risky and lucky actions which occasionally generate large but ephemeral short-term gains: it’s the disciplined management of risk which consistently avoids large and enduring losses. Most fundamentally, avoiding stupidity – and the huge losses and poor long-term returns it begets – is much less difficult than developing or finding brilliance.

Minimising losses and letting gains mind themselves, and developing a humble and stoic mindset, is far more productive than overconfidently striving to outperform. Don’t overstate returns: disregard average results and emphasise the most likely ones. On that basis, locate the handful of worthwhile investments by shunning the many mediocre – and worse – ones.

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