5.1 Expected Value Math

Replace "winning trades" thinking with "expected value" thinking. This single shift is what separates gamblers from professionals.

Layer 5: The Meta Game — Chapter 1 Goal: Replace "winning trades" thinking with "expected value" thinking. This single shift is what separates gamblers from professionals.

The Core Idea

You don't make money by being right. You make money by having positive expected value (EV) and managing variance.

A trader with 40% win rate can be highly profitable. A trader with 90% win rate can blow up their account. The math is what matters, not the feeling of being right.

This chapter is the most important in Layer 5. Internalize it.

The Expected Value Formula

Expected Value (EV) = (Win Rate × Average Win) - (Loss Rate × Average Loss)

Variables Defined

Win Rate: % of trades that are profitable
Average Win: average $ gain on winning trades
Loss Rate: % of trades that lose (= 1 - Win Rate)
Average Loss: average $ loss on losing trades

Example 1: A "Bad" Win Rate, Profitable System

Win rate: 40%
Average win: $1,000
Loss rate: 60%
Average loss: $400

EV = (0.40 × $1,000) - (0.60 × $400)
   = $400 - $240
   = $160 per trade

Positive EV. Over 100 trades, expected profit = $16,000.

Example 2: A "Good" Win Rate, Losing System

Win rate: 80%
Average win: $100
Loss rate: 20%
Average loss: $500

EV = (0.80 × $100) - (0.20 × $500)
   = $80 - $100
   = -$20 per trade

Negative EV. Over 100 trades, expected loss = $2,000. Despite "winning" 80% of the time.

The Lesson

Win rate alone is useless. Without knowing win/loss size, you can't tell if a strategy is profitable.

R-Multiples: The Better Way to Think

Instead of dollars, think in terms of R = the amount risked per trade.

The Mechanic

You enter a trade
Your stop is at a point where you'd lose $X
That $X is "1R" for this trade
If you win 2× your risk, you made 2R
If you lose your stop, you lost -1R

Why R-Multiples Are Better

Comparable across position sizes. A 2R win is 2R whether you risked $50 or $5,000.
Forces stop-loss thinking. You can't calculate R without a stop.
Account-size independent. Track performance in R, not dollars.
Reveals true strategy edge. Average R per trade is your edge.

Example

You risk $200 on a trade (your stop = -$200).

If you make $400 → 2R win
If you make $600 → 3R win
If you lose your stop → -1R
If stopped out at -$150 → -0.75R

Expectancy in R-Multiples

Expectancy = (Win Rate × Average Win R) - (Loss Rate × Average Loss R)

Examples

System A

Win rate: 50%
Average win: 2R
Average loss: -1R
Expectancy: (0.5 × 2R) - (0.5 × 1R) = +0.5R per trade

System B

Win rate: 30%
Average win: 4R
Average loss: -1R
Expectancy: (0.3 × 4R) - (0.7 × 1R) = +0.5R per trade

System C (Losing)

Win rate: 60%
Average win: 1R
Average loss: -2R
Expectancy: (0.6 × 1R) - (0.4 × 2R) = -0.2R per trade

Systems A and B are equally good despite very different win rates.

The Goal

Maximize expectancy in R. A system with +0.3R expectancy over 200 trades = 60R of profit. If your R = $200, that's $12,000.

Risk/Reward Ratio (R:R)

Definition

The ratio of potential reward to potential risk on a trade.

R:R = (Target - Entry) / (Entry - Stop)

Examples

Entry: $200, Stop: $195, Target: $210
R:R = ($210 - $200) / ($200 - $195) = $10 / $5 = 2:1 (2R reward, 1R risk)

Why R:R Matters

With a 50% win rate:

1:1 R:R → break-even
2:1 R:R → +0.5R expectancy
3:1 R:R → +1R expectancy

Minimum Acceptable R:R

For most swing trades, 2:1 R:R minimum. Below that, you need very high win rates.

Don't Force R:R

If a setup naturally has a wider stop or closer target, accept that. Adjust position size instead to keep $ risk constant.

The Win Rate vs. R:R Trade-Off

Win Rate	R:R Needed for Profit
80%	0.25:1 (1:4)
70%	0.43:1 (3:7)
60%	0.67:1 (2:3)
50%	1:1
40%	1.5:1
30%	2.33:1
20%	4:1
10%	9:1

What This Shows

High win rate strategies need only modest R:R
Low win rate strategies need high R:R
Most retail strategies aim for middle ground (50-60% win rate with 1.5-2.5 R:R)

Real Profitable Approaches

Day traders: ~50% win rate with 1.5-2 R:R
Swing traders: ~40-50% win rate with 2-3 R:R
Trend followers: ~30-40% win rate with 3-5 R:R
Scalpers: ~70-80% win rate with 0.5-1 R:R

All can be profitable. Pick a model that fits your psychology.

Profit Factor

Another useful metric.

Definition

Profit Factor = Gross Profit / Gross Loss

Examples

Total profits from winners: $10,000
Total losses from losers: $5,000
Profit Factor: 2.0

Interpretation

PF < 1.0: losing system
PF = 1.0: break-even
PF 1.0-1.5: marginal
PF 1.5-2.0: solid
PF > 2.0: very strong
PF > 3.0: elite (rare and often overfitted)

Caveat

A high PF doesn't mean low risk. A strategy with PF 2.0 but maximum drawdown of 60% is dangerous despite the math.

Sharpe Ratio (Risk-Adjusted Returns)

Definition

Sharpe Ratio = (Strategy Return - Risk-Free Rate) / Standard Deviation of Returns

In plain terms: returns per unit of risk.

Interpretation

< 0: worse than risk-free
0-1: mediocre
1-2: solid
2-3: very good
> 3: excellent (rare and often unsustainable)

Why It Matters

A strategy returning 100% per year with 80% drawdowns is worse than one returning 30% with 10% drawdowns.

Sharpe normalizes for risk.

Caveat

Sharpe penalizes upside volatility equally with downside. This isn't ideal — see Sortino.

Sortino Ratio

Like Sharpe, but only considers downside volatility.

Why It's Better

Upside swings shouldn't count as "risk." Sortino measures only downside risk.

Use

Strategies with positive skew (occasional big winners) often have better Sortino than Sharpe
More aligned with how traders actually feel about risk

Interpretation

Similar bands to Sharpe, but typically Sortino > Sharpe for positive-skew strategies.

Calmar Ratio

Calmar Ratio = Annualized Return / Max Drawdown

< 0.5: poor
0.5-1.0: acceptable
1.0-3.0: solid
> 3.0: strong

Tracking Your Expectancy

A Simple Spreadsheet

Trade #	Stock	Risk ($)	P/L ($)	R-Multiple
1	AMD	200	+400	+2.0 R
2	NVDA	200	-200	-1.0 R
3	MSFT	150	+75	+0.5 R
4	AAPL	250	+500	+2.0 R
5	TSLA	200	-200	-1.0 R

Running Stats

Win rate: 60% (3/5)
Average win R: (2 + 0.5 + 2) / 3 = 1.5R
Average loss R: 1R
Expectancy: (0.6 × 1.5) - (0.4 × 1) = 0.5R per trade

After 50-100 trades, this gives you real data about your edge.

The Sample Size Problem

You can't determine your true expectancy from 5 trades. Or 20. Or even 50.

Why

Variance dominates small samples
A bad streak in a good system happens by chance
A good streak in a bad system happens by chance
You need 100+ trades to start trusting the numbers

Practical

Track every trade
After 100 trades, review expectancy
After 200 trades, you can trust it more
Recognize that 30 trades tells you almost nothing

We'll cover this more in Chapter 5.4 (Variance and Sample Size).

Each flip is a trade
Heads = win, tails = loss
You're paid more for heads than you lose for tails (positive R:R)
The coin lands tails more often than heads (lower win rate)
Each flip is random, but over thousands of flips, you profit
A bad streak doesn't mean the coin is broken — it's just variance
A good streak doesn't mean you're a genius — also variance

Trading professionally means trusting the long-term math while surviving the short-term variance.

Building a Profitable System

The minimum requirements:

Positive expectancy in R. Even 0.3R is meaningful.
Defined R per trade. Always.
Strategy with edge in the current market regime.
Sample size discipline. Don't judge from 10 trades.
Variance survival. Position sizing protects through losing streaks.

This is what Layer 5 builds.

Practical Takeaways

Expected value is king. Win rate alone is meaningless.
R-multiples are the right unit for thinking about trades.
Minimum 2:1 R:R for most swing trades. Or maintain very high win rate.
Profit Factor > 1.5 is good. Above 2.0 is great.
Sharpe/Sortino normalize for risk. Use to compare strategies.
Sample size matters. 100+ trades before judging a strategy.
Track every trade in R. This is your edge measurement.

Quick Self-Check

Before moving to 5.2, you should be able to answer:

What's the expected value formula?
Can a 30% win rate strategy be profitable? How?
What is an R-multiple?
What's the minimum R:R for swing trades?
What's the difference between Sharpe and Sortino?
What does Profit Factor measure?
Why is 30 trades not enough to judge a strategy?

Next: 5.2 Risk of Ruin