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Speculation is to investing, what “cheat days” are to a diet regimen. In my experience having a bad meal every now and then makes you much more likely to stick to your diet. Similarly, by giving ourselves permission to speculate every now and then, we stand a better chance of remaining disciplined when making our “real investments”.
To be clear, when it comes to the stock market, I spend the majority of my time, energy, and capital on investing, not speculation. But I have been thinking about trading (aka speculation) a lot lately. And last week I was reading Roger Federer’s 2024 Dartmouth commencement speech where he revealed something very interesting.
Roger Federer, arguably the greatest tennis player ever, won 82% of his 1,500+ career matches.
But did you know…he won only 54% of the individual points he played.
Just 54%!
That’s barely better than a coin flip.
How does a mere 4% edge above a coin flip produce an 82% win rate?
The answer turns out to be one of the most powerful concepts in trading.
I call it The Federer Equation.
Federer’s Secret
Sure, attributes like clutch performance under pressure and mental toughness go a long way in building such an amazing track record, but ultimately the reason 54% becomes 82% comes down to math, and the structure of tennis scoring.
Tennis has a nested hierarchy:
Points build into Games (first to win 4 points, with deuce)
Games build into Sets (first to win 6 games, with tiebreak)
Sets build into Matches (best of 3 or best of 5)
At each level, a small edge gets amplified.
Here’s how.
From Points to Games
A tennis game works like this:
You need to win 4 points before your opponent does.
If it’s tied 3-3 (deuce), you need to win 2 in a row.
So Federer walked into every game with a 54% chance of winning each point (p = 0.54) and his opponent 46% (q = 0.46).
Let’s dust off some high school probability and combinatorics, and work through every way Federer could win a game.
Win 4-0 (a “love game”):
Federer wins 4 points in a row.
Probability = p⁴
Win 4-1:
Federer wins 4 points, his opponent wins exactly 1.
The opponent’s point can land in any of the first 4 positions (not the last since Federer must win the final point). That’s C(4,1) = 4 ways.
Probability = 4 × p⁴ × q¹ = 4p⁴q
Win 4-2:
Federer wins 4, opponent wins 2.
There are 10 ways this can happen (because the opponent’s 2 points can come in any of the first 5 points). That’s C(5,2) = 5 Choose 2 = 10.
Probability = 10 × p⁴ × q² = 10p⁴q²
Reach deuce (3-3), then win from deuce:
Both players win 3 of the first 6 points. That’s C(6,3) = 20 ways to reach deuce.
Once at deuce, Federer needs to win 2 points in a row. But if they split the next 2 points, it resets to deuce. This can repeat forever.
The probability of eventually winning from deuce is: p² / (p² + q²)
Therefore Probability = 20 × p³ × q³ × p² / (p² + q²) = 20p³q³ × p²/(p²+q²)
Adding all four scenarios together gives us:
P(game) = p⁴ + 4p⁴q + 10p⁴q² + 20p³q³ × p²/(p²+q²)
Plugging in p = 0.54 gives us
P(game) = 8.5% + 15.6% + 18.0% + 17.8% = 59.9% ≈ 60%
⇒ A 4% edge above 50/50 became a 10% edge!
How cool is that?!
From Games to Sets
A set works the same way, just one level up.
Instead of “first to 4 points,” it’s “first to 6 games.”
If it reaches 6-6, Federer needs to win 2 in a row.
To win a set, he needs to reach 6 games before his opponent.
The same pattern applies.
Win 6-0: 0.60⁶ = 4.7%
Win 6-1: C(6,1) × 0.60⁶ × 0.40¹ = 11.2%
Win 6-2: C(7,2) × 0.60⁶ × 0.40² = 15.7%
Math Explained: To win a set 6-2, Federer wins 6 games and his opponent wins exactly 2. Federer must win the last game (that’s what ends the set), so we only need to figure out the arrangement of the first 7 games.
In those first 7 games, Federer wins 5 and his opponent wins 2. The opponent’s 2 wins can land in any of those first 7 points. That gives us: C(7,2)
Win 6-3: C(8,3) × 0.60⁶ × 0.40³ = 16.7%
Win 6-4: C(9,4) × 0.60⁶ × 0.40⁴ = 15.0%
Reach 5-5, then win from there: C(10,5)*.65*.45*.69 = 13.9%
Math Explained: There are C(10,5) = 252 to get to 5-5.
Once at deuce, Federer needs to win 2 games in a row. But if they split the next 2 games, it resets to tie-break. This can repeat forever.
The probability of eventually winning from deuce is: P(game)² / (P(game)² + (1-P(game))²) = 0.69
P(set) = 4.7+11.2+15.7+16.7+15+13.9 = 77.2%
⇒ The jump from 60% to 77.2% is +17 percentage points, even larger than the +6 point jump from points to games!
From Sets to Matches
For simplicity, let’s assume Federer’s matches were best-of-3 sets.
To win, he needs 2 sets before his opponent gets 2.
With a 77.2% set win rate, there are only two winning scenarios:
Win 2-0: 0.772 × 0.772 = 59.6%
Win 2-1: The lost set can be either the 1st or 2nd (not the 3rd). That’s 2 arrangements: 2 × 0.772² × (1-.772) = 27.2%
The formula:
P(match, Bo3) = P(set)² + 2 × P(set)² × (1 - P(set))
⇒ P(match) = 59.6% + 27.2% = 86.8
The Full Cascade: How 54% cascades through every level
A 4% edge at the point level became a 36.8% edge at the match level!
Read that again.
Federer wasn’t 39% better than his opponents.
He was only 4% better.
But tennis structure, and more importantly, his consistency did the rest.
This is the Federer Equation:
A small, consistent edge at the unit level, repeated across many trials within a structured hierarchy, compounds into a dominant edge.
And it applies directly to trading.
Mapping The Federer Equation to Trading
The tennis hierarchy has a direct analog in trading: your trading edge doesn’t need to be large on any single trade. It needs to be consistent and contained.
The Equation Applied
Imagine a trader with the following daily stats:
The expected value of these daily stats is
EV = p × W - (1-p) × L = 0.6*1.2 - 0.4&1.25 = +0.22%
That’s just $22 per day on a $10,000 account.
The break-even win rate is the point where EV = 0.
EV = p × W - (1-p) × L = 0
Solving for p: Break-even = L / (W + L) = 1.25 / (1.20 + 1.25) = 51.0%
This trader’s win rate of 60% is 9 percentage points above break-even.
That’s the edge.
It’s modest, but it’s real.
At first glance, this doesn’t look like much.
But look what happens when we run the cascade.
The Weekly Cascade
Probability of a winning week: 68.26% (top 3 rows)
The Full Monthly Cascade
Probability of a winning month: A whopping 90.25% (top 3 rows)
A 60% daily win rate, becomes a 90.25% winning month rate!
And look at the W/L ratio transformation:
At the daily level, it’s 0.96x: losses are actually bigger than wins.
By the monthly level, it’s a whopping 1.27x: losses are no longer bigger than wins!
The cascade restructures your risk/reward profile, even when you start with nearly equal wins and losses!
$22 per day becomes a +74% cumulative annual projected return!
All from a 60% win rate and a razor-thin edge per day.
It turns out Federer wasn’t dramatically better than his opponents on any single point.
He was slightly better, consistently.
The Four Quadrants, Revisited
If you’ve been reading this Playing For Doubles for a while, you might recall that back in 2020, I wrote about the relationship between batting average (win rate) and W/L ratio in Many Bets, A Few Big Payoffs.
In that piece, I laid out four quadrants:
The key equation from that article was:
For a positive expected return: W/L > (1 - B) / B
where B = batting average (win rate)
Our theoretical trader lives in Quadrant A: High win rate, sub-1.0 W/L ratio.
Here’s what the Federer Equation adds to that earlier framework: it shows what happens to your edge over time.
The “Many Bets” article told you whether you have a positive EV on a per-trade basis.
The Federer Equation tells you how that per-trade EV compounds through layers of repetition.
Think of it this way:
“Many Bets” answers: Do I have an edge?
The Federer Equation answers: How big does that edge become over a week, a month, a year?
Note: The Federer Equation works on Quadrant B (VC Model) as well. The cascades just behave differently.
Build Your Own Federer Equation
I’ve built a free spreadsheet that lets you plug in your own numbers and see the cascade in action.
The spreadsheet shows you:
Whether you have an edge, based on your own statistics
Your break-even win rate and how far above it you are
The full cascade (how your win rate amplifies through the hierarchy)
The outcome distribution at each level
A 12-month projected return
Plug in your own numbers, and see where your edge comes from.
The Bottom Line
Roger Federer didn’t win 82% of his matches because he was 32% better than everyone else.
He won because he was 4% better, consistently, and the structure of repeated competition amplified that small edge into dominance.
The same math applies to trading:
You don’t need to be right on every trade.
Your edge can be small.
Consistency beats brilliance.
The next time you’re staring at a red day in your trading account, remember the Federer Equation.
Any given loss isn’t the problem.
The average loss growing beyond its designed size is.
Take the loss. Move on. Play the next point.
The cascade will do the rest.
Disclaimer: Not Financial Advice…of course
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