Understand risk of ruin concept

WHAT is Risk of Ruin?

Three levers control it:

• Win rate $W$ — fraction of trades that win.
• Reward-to-risk ratio $R$ — average win size divided by average loss size (a payoff descriptor, not the same as expectancy).
• Risk per trade $f$ — fraction of capital risked on each trade (position sizing).

The edge (expectancy per unit risked) is a derived quantity that combines the first two: $\text{Edge } G = W(R+1) - 1.$ $G>0$ means positive expectancy. The scary insight: you can have a profitable system ($G>0$) and still go broke if $f$ is too large. RoR is where edge meets survival.

WHY it exists — the survival problem

Expectancy tells you if you win on average over infinite trades. But you don't trade infinite times with infinite capital. Losses cluster. A run of losers early on can hit zero, and zero is an absorbing barrier — once you're out, the average never gets a chance to save you.

HOW to derive a simple RoR formula (from scratch)

Let's build the simplest version: fixed bet size, symmetric outcomes (you win or lose the same unit amount each trade).

Setup. Let $p$ = probability of winning one unit, $q = 1-p$ = probability of losing one unit. You start with $N$ units of capital. Ruin = reaching $0$ units.

Step 1 — define the unknown. Let $r(N)$ = probability of ruin starting from $N$ units.

Why this step? We want a function of current capital; ruin depends only on how many units cushion you have (memoryless).

Step 2 — one-step recursion. From $N$ units, the next trade either wins (→ $N+1$) or loses (→ $N-1$): $r(N) = p\cdot r(N+1) + q\cdot r(N-1)$

Why this step? Total probability: condition on the two possible next outcomes.

Step 3 — boundary conditions. $r(0)=1$ (already ruined), and $r(\infty)=0$ (with infinite capital you never go broke if edge is positive).

Step 4 — solve the recursion. Try $r(N)=x^N$. Substituting: $x = p\,x^2 + q \;\Rightarrow\; p\,x^2 - x + q = 0$ Factor (since $p+q=1$): roots $x=1$ and $x = q/p$.

Why this step? Linear recurrences have geometric solutions; the characteristic equation gives the growth ratio.

Step 5 — apply boundaries. The general solution is $r(N)=A + B(q/p)^N$. Using $r(\infty)=0$ (needs $q<p$ so the ratio $<1$) and $r(0)=1$:

Sanity checks (Forecast-then-Verify):

• If $p=q=0.5$ (no edge): $q/p=1 \Rightarrow r=1$ → ruin is certain. ✔ (a fair coin game with a floor at 0 always eventually hits 0)
• If $p>q$ (real edge): base $<1$, and raising it to power $N$ → smaller RoR as $N$ grows. More units of cushion = safer. ✔

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

Real trades have a reward-to-risk $R$ (average win / average loss) and you risk a fraction $f$ each time. There is no simple closed form for RoR under proportional (multiplicative) betting, so practitioners use a heuristic borrowed from the fixed-unit gambler's-ruin model. Treat it as an ordering guide, not a precise number:

The takeaway is structural, not the exact digit:

• ==Bigger edge $G$ → base shrinks → RoR crashes toward 0.==
• ==More units $U$ (smaller $f$) → exponent grows → RoR crashes toward 0.==

Worked Examples

Common Mistakes

Flashcards

Connections

• Position Sizing — controls $f$, the strongest RoR lever.
• Kelly Criterion — optimal $f$ that balances growth vs ruin.
• Expectancy & Edge — supplies $G = W(R+1)-1$, the base of the RoR heuristic.
• Reward-to-Risk Ratio — the $R$ that feeds into edge (but is not edge itself).
• Drawdown Management — the path to ruin; RoR is its worst outcome.
• Random Walk & Absorbing Barriers — the math engine behind the derivation.

Concept Map

Hinglish (regional understanding)