Are You Losing A Game You Should Win?
Good Bets that Destroy Portfolios: The Hidden Trap of Compounding
Why Do I Burn Capital When Risk/Reward Skews My Way?
[Accreditation: This post was inspired by a discussion with my friend Mauricio who writes under the pseudonym Polymath Investor. Thank you for the inspiration amigo!]
Let’s play a coin-flip game.
You start with a $100 of capital.
The coin is fair: an equal chance of landing on ‘heads’ or ‘tails’.
If you win, your capital will increase by 50%. If you lose, your capital reduces by 40%.
So after the first flip, you’ll either have $150 or $60. Up $50 or down $40.
Are you willing to play?
At first glance, it looks attractive. You make more on a win than you lose on a loss, while chances are 50:50.
Most people would say “yes”.
Most people would lose money.
What? Why?
It’s all about understanding markets, compounding, managing risk and some very simple math.
Russian Roulette Explains it Best
Russian Roulette is a ‘game of chance’ that involves placing a single bullet in one of six chambers of a revolver. The player spins the revolver (hence the ‘roulette’ reference), points the gun to his head, and pulls the trigger.
The player has five-in-six chance (83%) of survival, a win, and a corresponding 17% chance of losing (certain death).
The odds are favourable, but the risk is unacceptable. If you lose, there is no chance of ever making a recovery. Even if the incentive to play was a cool $1 billion US dollars if you survive, are you taking the chance?
As this demonstrates, what ought to determine your decision to play is not the size of the prize, but the extent of the possible loss and how difficult it will be to recover from it.
In the world of investing, this is the part that most people miss. They become captivated by upside potential while giving too little weight to downside risk, drawdowns, and the mathematics of recovery.
Perhaps that is because, in the stock market, it is not their life at stake. But financially, the consequences can still be equally severe.
LTCM: When Genius Ignores Path Dependency
Long-Term Capital Management (LTCM) was an investment firm established in 1994 and stacked with elite minds, including Nobel Prize winning mathematicians and economists.
Early returns were extraordinary.
Year 1 saw annualized returns of over 21% (after fees). Year two and three were even better at 43% and 41% respectively.
But in 1998, its fourth year of operation, it lost $4.6 billion. Inadvertently, it had been playing Russian Roulette in a financial context and it met an abrupt and sudden death.
Brilliance didn’t save them from mathematics.
So what can we learn from this?
Back to the Coin Flip
Be reminded that you begin with $100, if you win, your capital increases in value by 50%. If you lose, your capital reduces by 40%.
Let’s keep this simple and consider only two flips of the coin. There can only be four possible combinations from two flips:
Win/Win [CAGR 50%, two years, capital value = $225]
Win/Lose [+50% to $150, -40% to an end capital value = $90]
Lose/Win [-40% to $60, +50% to an end capital value = $90]
Lose/Lose [CAGR -40%, two years, capital value = $36]
Total of all outomes: $225 + $90 + $90 + $36 = $441
Average (mean) result: $441 / 4 = $110.25
If I start with $100, and the average outcome appears to be $110.25 after only two flips of the coin, that looks great, right?
Wrong!
Think about the median outcome. Three of the four paths leave you below where you started.
You have a 75% chance of losing.
That’s the trap.
Finance is full of this mistake: assuming that favourable averages for the market as a whole automatically translate into favourable long-term results for each individual investor.
The winning path, on which you have only a 25% probability of landing, delivers an outsized compounded return that pulls the overall average higher.
The gain from that single favourable path exceeds the combined losses from the other three possible paths, which is precisely why the arithmetic average appears skewed to the upside. But it’s a misleading signal. Do you want to accept a three-in-four chance of losing money?
It’s like Russian Roulette all over again, but this time with a modified chamber. Rather than only one bullet in six chambers of the revolver, you are playing with three bullets in only four chambers!
Do you still want to play?
Although a 50% upside versus a 40% downside on a 50:50 coin flip may look asymmetrically favourable, the real problem is that you are thinking in percentages rather than monetary outcomes.
Why does that matter?
If someone offered you $50 for a win while risking $40 for a loss, you would take that bet all day long. Over a large number of coin flips, heads and tails tend to balance out. So, for every $40 loss, you collect $50 on a win. That is a genuinely favourable skew.
But that is not what is happening when the outcomes are expressed as +50% and -40%.
Percentages are relative to the changing value of your capital. They do not offset each other in a straight-line way.
For example, a 50% loss requires a 100% gain just to get back to where you started. Likewise, if you gain 100% on one flip and then lose 50% on the next, you end up exactly where you began.
This is why percentage outcomes can be deceptive. A gain and a loss of similar size are not equal in economic terms.
So, if you are measuring returns in percentages, the upside must be materially larger than the downside for the odds to be truly skewed in your favour (see chart below):
The Basic Math
Let’s break this down in simple terms looking at the geometric mean as a proxy for median1. Below are four scenarios (A to D) that demonstrate why chasing large gains is what kills most investors portfolios and why both Howard Marks and Warren Buffett advise that the focus should be on managing downside risk, after which the upside will look after itself.
(Scenario A)
The example we considered above (Win: +50%, Loss: -40%):
Chance of a win = 0.5; Chance of a loss = 0.5
Impact on capital base of a win = 1.5x
Impact on capital base of a loss = 0.6x
Expected value per flip = (0.5×1.5) + (0.5×0.6) = 1.05
Since 1.05 > 1, we would expect a positive arithmetic outcome.
But the geometric mean is: √(1.5×0.6) = √0.9 = 0.949
Since 0.949 < 1, this reveals that the most probable outcome will be negative.
Essentially, we see negative geometric growth despite having a positive arithmetic expectation.
The arithmetic outcome is dominated by a vanishingly small number of exceptionally lucky sequences in which the player strings together multiple consecutive wins early on, allowing gains to compound strongly.
This is visible on a distribution chart through the pronounced long tail. Capital can only fall to zero, creating a hard floor on the downside, yet there remains a small probability of compounding upward to extremely large values.
That long right tail is what pulls the simple average higher and creates the misleading impression (a ‘false positive’ signal that the game offers a favourable expected return).
This is why people confuse positive expectancy with positive wealth creation.
They are not the same thing when compounding is involved.
It is the median outcome that represents what happens to the “typical” player, and what happens most often, is a loss of capital: a negative result.
As the number of flips increases, this situation becomes more pronounced; the median will slowly but surely pull your capital balance towards zero, while the mean will creep higher.
(Scenario B)
Digging deeper reveals that people chasing high returns, and risking larger drawdowns on their capital to do so, are on a fools errand (as LTCM learned the hard way).
Let’s keep the ratio of benefit to detriment unchanged, moving from +50:-40 in the last example, to +100:-80 (Win: +100%, Loss: -80%)
Chances of a win or loss are still 50:50
New impact on capital base of a win = 2.0x
New impact on capital base of a loss = 0.2x
Expected value per flip: (0.5×2.0) + (0.5×0.2) = 1.1 (the positive expectation has increased due to the higher win percentage lengthening the tail)
But the outcome has become way more volatile. Bigger win vs. bigger loss potential.
The geometric mean now: √(2.0 × 0.2) = √0.4 = 0.632
The geometric mean has fallen dramatically, and this remains the most likely outcome. This is a super dangerous game to play.
(Scenarios C and D)
So instead of chasing outsized returns, what happens if we focus on mitigating the downside (managing risk).
Again, we’ll keep the win:loss ratio the same, but take it in the other direction. Our initial +50:-40 first becomes +25:-20, and then +12.5:-10.
(Win: +25%, Loss: -20%)
Expected value per flip: 0.5×1.25 + 0.5×0.8 = 1.025
Geometric mean: √(1.25×0.8) = √1.0 = 1.0
A geometric mean of 1.0 is the breakeven point: zero growth and zero loss.
Now take a look at the ‘break-even’ bar chart above. You’ll see that +25% and -20% are indeed the balancing point.
So, we’ve moved from two guaranteed loss scenarios to break-even. Now we seem to be travelling in the correct direction.
If we reduce the loss percentage still further, are we able to push into a guaranteed sinning strategy?
(Win: +12.5%, Loss: -10%)
Expected value per flip: 0.5×1.125 + 0.5×0.9 = 1.0125
Geometric mean: √(1.125×0.9) = √1.0125 ≈ 1.0062
Positive geometric growth. This is the only scenario of the four studied with consistent long-term growth!
It becomes evident that the game is not won by maximising upside.
It is won by controlling downside.
Large drawdowns damage compounding more than large gains help it.
That’s why risk management matters more than return targets.
Kelly Gets It: The Magic Formula
The Kelly Criterion, popularised by J. L. Kelly and later championed by Ed Thorp, grew out of Claude Shannon’s Information Theory. At its core, it is a framework for maximising long-term compound growth.
Its most important lesson is simple: never risk ruin. Don’t bet the farm, no matter how attractive the opportunity appears. Survival comes first. This is where the Russian Roulette analogy sits.
So, Kelly’s central question was straightforward:
“What fraction of my capital should I allocate to the next opportunity, based on the probability of success and the payoff if I am right, relative to the loss if I am wrong?”
That works neatly in games of chance. A coin toss has known odds. A roulette wheel has fixed probabilities. The variables are measurable.
Equities are different.
When buying a stock, there is no objective probability of success. No formula can tell us the precise odds that a company will outperform, rerate, or compound over time. At best, we are making informed judgements. That makes the pure Kelly formula more elegant in theory than useful in practice for investors.
More particularly, if the win vs. loss payout skew is unfavourable, why stake any capital at all? Better opportunities exist elsewhere. In that sense, Kelly appears to be about forcing a bet and sizing it accordingly, which is something a rational investor would never do.
So, stepping out of the world of academia, how can investors use a derivative of the Kelly principle when investing?
Rather than pretending to know precise probabilities, we build scenarios.
Before investing, we develop a bull case and a bear case. In each scenario, we estimate what the business may be worth relative to today’s share price. From that, we can calculate:
the potential percentage gain if the thesis plays out
the potential percentage loss if it doesn’t
Those are practical, investable variables.
And once we have them, we can begin to think the Kelly way.
For algebraic purposes, we’ll call a win ‘X’ and a loss ‘Y’. For a 50/50 game, the breakeven condition for geometric growth is:
(1+X)(1-Y) = 1
Which may be rearranged to define X in terms of Y at breakeven, as follows:
1 - Y + X - XY = 1
X - Y - XY = 0
X(1-Y) = Y
X = Y/(1-Y)
Now we can suplant the numbers in our scenarios.
In scenario C, when the drawdown on a loss (Y)=20%: to achieve breakeven X would need to be 25% (20%/80%), which was exactly the win percentage, hence the breakeven outcome.
The only scenario that yields positive geometric growth is D, when the drawdown on a loss (Y)=10%. To achieve breakeven X would need to be ≈11.11% (10%/90%). Since the gain on a win was 12.5% (greater than 11.11%), the typical player actually makes money over time.
In short, to win, you need X > Y/(1-Y)
In the world of investing, this explains why volatility drag is real, why loss ratios matter, and why the understanding risk is essential for long-term growth.
Moral of the story
Many investors are playing financial Russian roulette.
They see favourable averages and ignore path dependency, volatility, and downside risk.
That’s how capital gets destroyed.
As Howard Marks always counsels, ‘Look after the downside and the upside will look after itself’.
Always focus on the risk!
Further Reading: More ‘Risk’ Content
If you enjoyed this, you will also like these:
Geometric Mean: Calculates the “central tendency” using multiplication, often used for percentages, ratios, and skewed data (e.g., population growth or investment returns).
Median: Simply the middle value, making it more robust against extreme outliers compared to any mean.
In perfectly symmetric log-normal data, the geometric mean and median are identical.
Hey James, I have to correct you. Most people would not play that game. How do I know? Because it's the result of a previous psychology study (I don't have the reference but it's a fairly well known phenomenon). It's called loss aversion. The interesting thing about the loss aversion is that people decide not to bet or miss the optimum gain even when they are offered multiple coin tosses and positive geometrical returns.
Also, there is an important point you do not mention. Geometrical returns only apply when you can only put all of your eggs into the same basket. You can calculate the geometrical returns of your whole portfolio and try to maximize that, but when you have multiple stocks or different asset classes, you need to use the arithmetical average to calculate the return for each time period. The Kelly bet is an extension of that: sizing your bet is like having a mix of cash and one stock, and the volatility for cash is 0, but the volatility of that stock is much higher. So having an allocation to something with a neutral or even negative arithmetical return (like buying puts) can improve geometrical returns. A good book to understand this is “Safe Haven” from Mark Spitznagel.
The problem here is that on a vacuum, what this tells us is that diversification is always better than concentration because it allows you to transform geometrical into arithmetical returns. But as value investors, we do not see all investments as having equal probabilities of success or equal pay offs. Then, we need to concentrate into our best ideas, contrary to what the mathematics of geometrical returns would tell us. On the opposite end of the spectrum, there are “academic” style portfolios that aim to maximize returns by using modern portfolio theory, and they use asset classes and strategies you have probably never heard of like trend following or managed futures, and lots of leverage (but very diversified). An example of that is what they call “return stacking”, which means using swaps to lever up two or more asset classes which are supposed to go in opposite directions, like stocks and bonds. The problem is that even though the strategies sound logical, when you look at the results of these funds, using complicated mathematics to determine what to buy and what to sell, results are underwhelming to say the least.
There are also a lot of differences between the nature of a coin toss and the stock market. If the stock market is a random walk, the longer you play, the more risk you are assuming. Value investors do not see it that way. I am concentrated into not only the best ideas I have, but the ideas I know will achieve very good returns even if I need to wait for 10 years to see it. That is the fundamental difference between risk and volatility. Many people see the headline, accounting earnings, or the price chart, while I am looking at the earning power and moat of the business. That shows a clear way of sizing bets: when time is on your side and you can achieve a superior result, you need to bet strongly. When you are speculating on events, even if the reward is really high, you need to bet less. Example to that is a company trading below cash but with weak management, compared to a company trading at low valuations but with high returns on invested capital and a long runway to grow. This also highlights a key trait to be a successful investor: you need to be persistent, rather than consistent.
Finally, another fundamental difference between the coin toss and the stock market is the nature of the distribution. Coin toss returns follow a log normal distribution. The stock market returns are distributed into a shape that resembles more a Lorentzian function. If you don't know what that is, its basically a more skewed version of the normal distribution.
Fantastic article to begin the week with!