Understand risk of ruin concept

4.7.9 · Stock-Market › Risk & Money Management

WHAT is Risk of Ruin?

Isse teen levers control karti hain:

• Win rate $W$ — trades ka wo fraction jo win hote hain.
• Reward-to-risk ratio $R$ — average win size divided by average loss size (ek payoff descriptor, not the same as expectancy).
• Risk per trade $f$ — har trade par capital ka wo fraction jo risk kiya jaata hai (position sizing).

Edge (expectancy per unit risked) ek derived quantity hai jo pehli do cheezein combine karti hai: $\text{Edge }G=W(R+1)-1.$ $G>0$ matlab positive expectancy. Ek darpok insight: tumhara system profitable ho sakta hai ($G>0$) aur phir bhi tum broke ho sakte ho agar $f$ bahut bada ho. RoR wahan hai jahan edge aur survival milte hain.

WHY it exists — the survival problem

Expectancy batata hai ki tum average par infinite trades ke baad jeette ho ya nahi. Lekin tum infinite baar infinite capital ke saath trade nahi karte. Losses cluster karte hain. Shuru mein losers ki ek run zero tak pohonch sakti hai, aur zero ek absorbing barrier hai — ek baar bahar ho gaye, to average tumhe bachane ka mauka nahi paata.

HOW to derive a simple RoR formula (from scratch)

Chalte hain sabse simple version banate hain: fixed bet size, symmetric outcomes (tum har trade mein same unit amount win ya lose karte ho).

Setup. Maano $p$ = ek unit jeetnay ki probability, $q=1-p$ = ek unit haarne ki probability. Tum $N$ units of capital se start karte ho. Ruin = $0$ units tak pohonchna.

Step 1 — unknown define karo. Maano $r(N)$ = $N$ units se start karke ruin ki probability.

Yeh step kyun? Hume current capital ki ek function chahiye; ruin sirf is par depend karti hai ki tumhare paas kitne units ka cushion hai (memoryless).

Step 2 — one-step recursion. $N$ units se, agli trade ya to jeeti (→ $N+1$) ya haari (→ $N-1$): $r(N)=p\cdot r(N+1)+q\cdot r(N-1)$

Yeh step kyun? Total probability: do possible next outcomes par condition karo.

Step 3 — boundary conditions. $r(0)=1$ (already ruined), aur $r(\infty )=0$ (infinite capital ke saath tum kabhi broke nahi hote agar edge positive ho).

Step 4 — recursion solve karo. $r(N)=x^N$ try karo. Substitute karne par: $x=p{x}^2+q\Rightarrow p{x}^2-x+q=0$ Factor karo (kyunki $p+q=1$): roots $x=1$ aur $x=q/p$.

Yeh step kyun? Linear recurrences ke geometric solutions hote hain; characteristic equation growth ratio deti hai.

Step 5 — boundaries apply karo. General solution hai $r(N)=A+B(q/p{)}^N$. $r(\infty )=0$ use karte hue (chahiye $q<p$ taaki ratio $<1$ ho) aur $r(0)=1$:

Sanity checks (Forecast-then-Verify):

• Agar $p=q=0.5$ (koi edge nahi): $q/p=1\Rightarrow r=1$ → ruin certain hai. ✔ (ek fair coin game mein floor 0 par hone se hamesha 0 hit hoga)
• Agar $p>q$ (real edge): base $<1$, aur isse power $N$ tak uthane par → RoR chota hota jaata hai jaise $N$ badhta hai. Zyada units of cushion = zyada safe. ✔

The practical version (unequal win/loss & fractional risk)

Real trades mein ek reward-to-risk $R$ (average win / average loss) hota hai aur tum har baar ek fraction $f$ risk karte ho. Proportional (multiplicative) betting ke liye RoR ka koi simple closed form nahi hai, isliye practitioners ek heuristic use karte hain jo fixed-unit gambler's-ruin model se liya gaya hai. Isse ek ordering guide ki tarah treat karo, precise number ki tarah nahi:

Takeaway structural hai, exact digit nahi:

• ==Bada edge $G$ → base shrinks → RoR crashes toward 0.==
• ==Zyada units $U$ (chota $f$) → exponent grows → RoR crashes toward 0.==

Worked Examples

Common Mistakes

Flashcards

Connections

• Position Sizing — $f$ control karta hai, sabse strong RoR lever.
• Kelly Criterion — optimal $f$ jo growth aur ruin ke beech balance karta hai.
• Expectancy & Edge — $G=W(R+1)-1$ supply karta hai, RoR heuristic ka base.
• Reward-to-Risk Ratio — woh $R$ jo edge mein jaata hai (lekin edge nahi hai).
• Drawdown Management — ruin ka path; RoR uska worst outcome hai.
• Random Walk & Absorbing Barriers — derivation ke peeche ka math engine.

Concept Map