Understand the Kelly Criterion — Hinglish notes

4.7.5 · Stock-Market › Risk & Money Management

YE EXIST KYU KARTA HAI?

YE KAUNSA problem solve karta hai? Tumhare paas ek edge hai (ek positive-expectation bet ya trade). Sawaal: kitna capital har baar risk karo?

Bahut zyada bet → ek buri streak tumhe barbaad kar deti hai (ruin permanent hai — $0 se recover nahi ho sakta).
Bahut kam bet → tumhara paisa dard-naak dheere badhta hai.

SIRF "expected value maximize" kyu nahi karte? Seedhe socho toh, agar ek bet ka positive EV hai, toh EV-maximizing kehta hai har baar sab kuch bet karo. Lekin isse eventual ruin pakki hai: apni wealth ko kai factors se multiply karo aur ek bhi $0$ factor poora product $0$ kar deta hai hamesha ke liye.

SCRATCH SE DERIVE KAISE KARTE HAIN (simple bet case)

Setup: ek bet jisme win par tumhare stake ka $b$ times milta hai (net odds $b$ -to-1), win probability $p$ , lose probability $q = 1 - p$ (haar gaye toh stake gaya).

Maano $f$ = har round mein bet kiya gaya bankroll ka fraction. Ek round ke baad, wealth multiply hoti hai:

Win: (1 + b f), Lose: (1 - f)

Multiply kyu? Kyunki agले round mein tum apne naye bankroll ka fraction bet karte ho — ye compound hota hai.

$N$ rounds ke baad jisme $W$ wins aur $L = N - W$ losses hain:

X_{N} = X_{0} (1 + b f)^{W} (1 - f)^{L}

Exponential growth rate per round define karo:

g (f) = \frac{1}{N} ln \frac{X _{N}}{X _{0}} = \frac{W}{N} ln (1 + b f) + \frac{L}{N} ln (1 - f)

Log kyu? Taaki growth rate ek sum (ek average) ban jaaye, jo Law of Large Numbers ki wajah se $N \to \infty$ par apni expectation par converge karta hai:

g (f) = p ln (1 + b f) + q ln (1 - f)

Ab $g (f)$ maximize karo — derivative lo aur zero ke barabar rakho. Ye step kyu? Optimal growth wahan hai jahan $g$ ki slope flat ho.

g^{'} (f) = \frac{p b}{1 + b f} - \frac{q}{1 - f} = 0

Solve karo: $p b (1 - f) = q (1 + b f)$

p b - p b f = q + q b f \Rightarrow p b - q = q b f + p b f = b f (p + q) = b f

Kyunki $p + q = 1$ :

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

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

Agar win sirf tumhara stake double kare ( $b = 1$ ):

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

Toh $p = 0.55$ ke saath: $f^{*} = 0.10$ → bankroll ka 10% bet karo. Intuitive aur simple: apni edge ka double bet karo.

Continuous / trading version

Returns ke liye jahan risk-free ke upar expected excess return $μ$ ho, aur variance $σ^{2}$ ho, log-growth maximize karne par milta hai:

f^{*} = \frac{μ}{σ ^{2}}

Ye form kyu? Normally-distributed return par log-utility: $g (f) \approx f μ - \frac{1}{2} f^{2} σ^{2}$ . Differentiate karo: $g^{'} (f) = μ - f σ^{2} = 0 \Rightarrow f^{*} = μ / σ^{2}$ . Isliye Kelly $\approx$ leverage proportional to Sharpe ratio.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

Kelly Criterion kya optimize karta hai?

Wealth ki long-run geometric growth rate, yaani ye

E [ln X]

maximize karta hai,

E [X]

nahi.

Win/loss bet ke liye Kelly fraction formula

f^{*} = \frac{p b - q}{b} = p - \frac{q}{b}

, jahan

p

=win prob,

q = 1 - p

b

=net odds.

Wealth ke log ko maximize kyu karte hain wealth ki jagah?

Wealth multiplicatively compound hoti hai; log products ko sums mein badalta hai isliye Law of Large Numbers guarantee karta hai ki average growth rate achieve hogi, aur ye ruin ko penalize karta hai.

Even-money Kelly fraction

f^{*} = 2 p - 1

(apni edge ka double bet karo) jab

b = 1

Trading ke liye continuous Kelly

f^{*} = μ / σ^{2}

(expected excess return over variance).

Kelly mein

b

ka matlab kya hai?

NET odds = staked par profit per unit, gross return nahi jisme tumhara stake bhi shamil ho.

Kaun se bet fraction par long-run growth zero cross karti hai (ruin)?

Even-money case (

b = 1

) mein exactly

f_{0} = 2 f^{*}

par; general odds ke liye ye

p ln (1 + b f) + q ln (1 - f) = 0

ka doosra root hai, generally

2 f^{*}

nahi.

Pros fractional (half) Kelly kyu use karte hain?

Ye zyaatar growth (~75%) rakhta hai jabki volatility/drawdowns roughly half kar deta hai, aur tumhara true edge over-estimate hone par hedge karta hai.

Agar

p b - q < 0

ho, toh Kelly kya kehta hai?

Kuch mat bet karo (ya doosri side);

f^{*} < 0

matlab koi positive-edge bet exist nahi karti.

Recall Feynman: 12-saal ke bacche ko explain karo

Socho tumhare paas ek magic coin hai jo tumhare favor mein zyada baar aaता hai, isliye uss par bet karna smart hai. Lekin trap ye hai: agar tum apna SAARA paisa bet karo aur ek baar bhi haro, toh tumhare paas zero hai — aur zero times kuch bhi zero hi rehta hai, isliye tum kabhi wapas nahi aa sakte. Kelly Criterion ek rule hai jo kehta hai "itna bet karo — na zyada, na kam." Agar tumhara coin 60% baar jeet ta hai aur bet double karta hai, toh Kelly kehta hai har baar apna 20% paisa risk karo. Iss tarah tumhara dher sabse tezi se badhta hai bina kabhi blow up hone ke risk ke. Ye isliye kaam karta hai kyunki tumhara paisa har round multiply hota hai, aur ek chhoti sankhya se multiply karna disaster hai, isliye tum wipeouts se bachte ho.

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