Understand the Kelly Criterion — Stock-Market notes

WHY does this even exist?

WHAT problem does it solve? You have an edge (a positive-expectation bet or trade). Question: how much of your capital do you risk each time?

Bet too much → one bad streak wipes you out (ruin is permanent — you can't recover from $0).
Bet too little → your money grows painfully slowly.

WHY not just "maximize expected value"? Naively, if a bet has positive EV, EV-maximizing says bet everything every time. But that guarantees eventual ruin: multiply your wealth by many factors and a single $0$ factor makes the product $0$ forever.

HOW to derive it from scratch (simple bet case)

Setup: a bet that on win returns $b$ times your stake (net odds $b$ -to-1), with win probability $p$ , lose probability $q = 1-p$ (you lose your stake).

Let $f$ = fraction of bankroll wagered each round. After one round, wealth multiplies by:

\text{Win: } (1 + bf), \qquad \text{Lose: } (1 - f)

Why multiply? Because next round you bet a fraction of your new bankroll — it compounds.

After $N$ rounds with $W$ wins and $L = N-W$ losses:

X_N = X_0 \,(1+bf)^{W}(1-f)^{L}

Define the exponential growth rate per round:

g(f) = \frac{1}{N}\ln\frac{X_N}{X_0} = \frac{W}{N}\ln(1+bf) + \frac{L}{N}\ln(1-f)

Why the log? So the growth rate is a sum (an average), which by the Law of Large Numbers converges to its expectation as $N \to \infty$ :

g(f) = p\,\ln(1+bf) + q\,\ln(1-f)

Now maximize $g(f)$ — take the derivative and set to zero. Why this step? The optimum growth is where the slope of $g$ is flat.

g'(f) = \frac{pb}{1+bf} - \frac{q}{1-f} = 0

Solve: $pb(1-f) = q(1+bf)$

pb - pbf = q + qbf \;\Rightarrow\; pb - q = qbf + pbf = bf(p+q) = bf

Since $p+q=1$ :

\boxed{\,f^* = \frac{pb - q}{b} = \frac{p(b+1)-1}{b}\,}

Even-money special case ( $b=1$ )

If a win just doubles your stake ( $b=1$ ):

f^* = \frac{p(1) - q}{1} = p - q = 2p - 1

So with $p=0.55$ : $f^* = 0.10$ → bet 10% of bankroll. Nice and intuitive: bet twice your edge.

Continuous / trading version

For returns with expected excess return $\mu$ over risk-free, and variance $\sigma^2$ , maximizing log-growth gives:

\boxed{\,f^* = \frac{\mu}{\sigma^2}\,}

Why this form? Log-utility on a normally-distributed return: $g(f) \approx f\mu - \tfrac{1}{2}f^2\sigma^2$ . Differentiate: $g'(f) = \mu - f\sigma^2 = 0 \Rightarrow f^* = \mu/\sigma^2$ . This is why Kelly $\approx$ leverage proportional to Sharpe ratio.

Worked Examples

Common Mistakes (Steel-manned)

Common mistake "Positive EV means bet as much as possible."

Why it feels right: Each bet has positive expected value, so more exposure = more expected profit, right? The flaw: That maximizes arithmetic mean, but wealth compounds geometrically. Over-betting drives the geometric growth rate negative even with positive EV. Beyond $f^*$ , growth actually falls, and past some fraction $f_0 > f^*$ the growth rate goes negative → ruin. In the even-money case ( $b=1$ ) that break-even fraction is exactly $f_0 = 2f^*$ ; for general odds $b$ , $f_0$ is the other root of $g(f)=0$ and is not $2f^*$ in general — it's where $p\ln(1+bf)+q\ln(1-f)=0$ . Fix: Maximize $E[\ln X]$ , not $E[X]$ . Use $f^*$ .

Flashcards

What does the Kelly Criterion optimize?

The long-run geometric growth rate of wealth, i.e. it maximizes

E[\ln X]

, not

E[X]

Kelly fraction formula for a win/loss bet

f^* = \dfrac{pb - q}{b} = p - \dfrac{q}{b}

, where

p

=win prob,

q=1-p

b

=net odds.

Why maximize log of wealth instead of wealth?

Wealth compounds multiplicatively; log turns products into sums so the Law of Large Numbers guarantees the average growth rate is achieved, and it penalizes ruin.

Even-money Kelly fraction

f^* = 2p - 1

(bet twice your edge) when

b=1

Continuous Kelly for trading

f^* = \mu/\sigma^2

(expected excess return over variance).

What does

b

mean in Kelly?

NET odds = profit per unit staked, not the gross return including your stake.

At what bet fraction does long-run growth cross zero (ruin)?

In the even-money case (

b=1

) at exactly

f_0=2f^*

; for general odds it is the other root of

p\ln(1+bf)+q\ln(1-f)=0

, generally not

2f^*

Why do pros use fractional (half) Kelly?

It keeps most of the growth (~75%) while roughly halving volatility/drawdowns, and hedges against over-estimating your true edge.

pb - q < 0

, what does Kelly say?

Bet nothing (or the other side);

f^*<0

means no positive-edge bet exists.

Recall Feynman: explain to a 12-year-old

Imagine you have a magic coin that lands your way more often than not, so betting on it is smart. But here's the trap: if you bet ALL your money and lose even once, you have zero — and zero times anything is still zero, so you can never come back. The Kelly Criterion is a rule that says "bet this much — not too much, not too little." If your coin wins 60% of the time and doubles your bet, Kelly says risk 20% of your money each time. That way your pile grows the fastest without ever risking blowing up. It works because your money multiplies each round, and multiplying by a small number is a disaster, so you must protect against wipeouts.

Connections

Risk & Money Management
Position Sizing
Expected Value vs Geometric Mean
Sharpe Ratio (Kelly ≈ Sharpe-proportional leverage)
Risk of Ruin
Volatility Drag
Logarithmic Utility
Compounding

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Kelly Criterion ka funda ekdum simple hai: agar tumhare paas ek edge hai (matlab bet ya trade jisme expected value positive hai), toh sawaal yeh nahi ki bet karna hai ya nahi — sawaal yeh hai ki kitna paisa lagana hai. Zyada laga diya toh ek bura streak aakar tumhara account zero kar dega, aur zero se wapas aana impossible hai (kyunki paisa multiply hota hai, add nahi). Thoda laga diya toh growth slow rahegi. Kelly exact sweet spot batata hai.

Formula yaad rakho: $f^* = \dfrac{pb - q}{b}$ , yaani "edge divided by odds". Yaha $p$ jeetne ka probability, $q=1-p$ haarne ka, aur $b$ net odds (jitna profit per 1 rupee lagaya). Simple even-money case mein toh aur easy: $f^* = 2p - 1$ — apne edge ka double bet karo. Jaise 60% win rate pe 20% bankroll lagao.

Yeh kaam kyun karta hai? Kyunki wealth har round mein multiply hoti hai. Isliye hum wealth ka nahi, wealth ke log ka average maximize karte hain — log products ko sum bana deta hai, aur sum pe Law of Large Numbers apply hota hai. Bas isi maths se optimal fraction nikalta hai. Ek baat dhyan rakho: even-money (b=1) case mein $2f^*$ pe growth zero ho jaata hai, par general odds mein yeh point alag hota hai — bas $g(f)=0$ ka doosra root nikalna padta hai.

Ek zaroori real-world tip: Full Kelly bahut aggressive hota hai — drawdowns bade aate hain, aur agar tumhara $p$ ka estimate galat hai toh tum actually over-bet kar rahe ho. Isliye pros Half Kelly ya Quarter Kelly use karte hain — growth ka 75% mil jaata hai par volatility aadhi ho jaati hai. Safe khelo, compound hone do.

Test yourself — Risk & Money Management

Connections