mahmoud20138 Skill: Risk of Ruin & Bankroll Mathematics | Domain: trading/risk-and-portfolio | Category: risk | Level: advanced Tags:
ruin,kelly,monte-carlo,bankroll,survival,sizing
Risk of Ruin & Bankroll Mathematics
1. Risk of Ruin Formula
Classical Formula
For a system with:
Win rate: W
Loss rate: L = 1 - W
Win/Loss ratio: R = avg_win / avg_loss
Risk per trade: f (fraction of bankroll)
Risk of Ruin = ((1 - Edge) / (1 + Edge))^(Capital_Units)
Where:
Edge = W × R - L = W × R - (1 - W)
Capital_Units = Account / Risk_Per_Trade
Practical Table
Win Rate: 55% | Win/Loss Ratio: 1.5:1 | Edge = 0.55×1.5 - 0.45 = 0.375
Risk/Trade Risk of Ruin Account Units
────────── ──────────── ─────────────
1% 0.001% 100 units
2% 0.1% 50 units
3% 1.2% 33 units
5% 12.4% 20 units
10% 52.1% 10 units
20% 89.3% 5 units
Win Rate: 45% | Win/Loss Ratio: 2.0:1 | Edge = 0.45×2.0 - 0.55 = 0.35
Risk/Trade Risk of Ruin
────────── ────────────
1% 0.01%
2% 0.3%
5% 18.7%
10% 61.2%
Key insight: Even with a positive edge, risking >5% per trade gives unacceptable ruin probability.
2. Kelly Criterion
Full Kelly
f* = (W × R - L) / R = Edge / R
Where:
f* = optimal fraction of bankroll to risk
W = win probability
R = win/loss ratio
L = 1 - W
Example:
W = 55%, R = 1.5
f* = (0.55 × 1.5 - 0.45) / 1.5
f* = 0.375 / 1.5
f* = 25% ← Full Kelly
Why Full Kelly Is Dangerous
Full Kelly:
✓ Maximizes long-term growth rate
✗ Produces stomach-churning drawdowns (50-85% is normal)
✗ Assumes known exact edge (you don't know this)
✗ Assumes infinite time horizon
✗ One estimation error can be fatal
RULE: Use fractional Kelly
Half Kelly (f*/2):
75% of the growth rate of full Kelly
Dramatically lower drawdowns
Much more robust to edge estimation errors
Quarter Kelly (f*/4):
50% of growth rate
Very smooth equity curve
Recommended for uncertain edge
Practical Kelly Sizing
def kelly_size(win_rate: float, avg_win: float, avg_loss: float,
kelly_fraction: float = 0.25) -> float:
"""
Quarter Kelly by default. Never go above half Kelly.
Returns: fraction of account to risk per trade
"""
R = avg_win / abs(avg_loss)
edge = win_rate * R - (1 - win_rate)
if edge <= 0:
return 0.0 # No edge = no trade
full_kelly = edge / R
return full_kelly * kelly_fraction
# Example:
# kelly_size(0.55, 150, 100, kelly_fraction=0.25)
# → Full Kelly = 25%, Quarter Kelly = 6.25%
# → Risk 6.25% of account per trade
#
# But cap at 2% maximum regardless of Kelly output
3. Monte Carlo Survival Analysis
Simulation Framework
import numpy as np
def monte_carlo_survival(
win_rate: float,
avg_win_r: float, # in R-multiples
avg_loss_r: float, # typically -1R
risk_per_trade: float, # fraction of current bankroll
num_trades: int = 500,
num_sims: int = 10000,
ruin_threshold: float = 0.5, # 50% drawdown = ruin
) -> dict:
"""
Simulate N trading paths and measure survival.
"""
ruin_count = 0
max_dds = []
final_balances = []
for _ in range(num_sims):
balance = 1.0 # normalized
peak = 1.0
max_dd = 0.0
ruined = False
for _ in range(num_trades):
if np.random.random() < win_rate:
pnl = balance * risk_per_trade * avg_win_r
else:
pnl = balance * risk_per_trade * avg_loss_r
balance += pnl
peak = max(peak, balance)
dd = (peak - balance) / peak
max_dd = max(max_dd, dd)
if balance <= (1.0 - ruin_threshold):
ruined = True
ruin_count += 1
break
max_dds.append(max_dd)
final_balances.append(balance)
return {
'ruin_probability': ruin_count / num_sims,
'median_max_dd': np.median(max_dds),
'p95_max_dd': np.percentile(max_dds, 95),
'median_final_balance': np.median(final_balances),
'p5_final_balance': np.percentile(final_balances, 5),
'p95_final_balance': np.percentile(final_balances, 95),
}
# Run with YOUR actual stats:
# result = monte_carlo_survival(
# win_rate=0.52, avg_win_r=1.8, avg_loss_r=-1.0,
# risk_per_trade=0.02, num_trades=500
# )
Interpreting Results
ACCEPTABLE:
Ruin probability: < 1%
P95 max drawdown: < 30%
P5 final balance: > starting balance
Median final balance: significantly above starting
WARNING:
Ruin probability: 1-5%
P95 max drawdown: 30-50%
→ Reduce risk_per_trade
UNACCEPTABLE:
Ruin probability: > 5%
→ System is not viable at this risk level
→ Either improve edge or reduce risk until ruin < 1%
4. Position Sizing Models
Fixed Fractional
Risk per trade = Account × Fixed_Percent
Pro: Simple, anti-martingale (bet less after losses)
Con: Can be slow to grow small accounts
Best for: Most traders, most of the time
Typical: 1-2% per trade
Volatility-Adjusted (ATR-Based)
Position Size = (Account × Risk%) / (ATR × ATR_Multiple)
Pro: Automatically adjusts for market volatility
Con: Requires reliable ATR calculation
Best for: Multi-instrument portfolios
Example:
Account: $100,000
Risk: 1% = $1,000
EURUSD ATR(14): 80 pips
ATR Multiple: 2 (stop at 2× ATR = 160 pips)
Position: $1,000 / 160 pips = 6.25 per pip ≈ 0.6 lots
Fixed Ratio (Ryan Jones)
Next level increase when: Profits ≥ Delta × Current_Units
Delta = chosen profit threshold per unit increase
Lower delta = more aggressive scaling
Higher delta = more conservative scaling
Pro: Geometric growth potential
Con: Complex, requires tracking
Best for: Scaling up proven strategies
5. Survival Rules
1. NEVER risk more than 2% per trade. Period.
2. NEVER have more than 6% total open risk.
(3 positions × 2% each, or 6 × 1%)
3. DAILY loss limit: 4% of account.
Hit it → done for the day. No exceptions.
4. WEEKLY loss limit: 8% of account.
Hit it → paper trade rest of week.
5. MONTHLY loss limit: 12% of account.
Hit it → full stop, review everything.
6. NEVER add to a losing position.
Average down = accelerate ruin.
7. Know your edge BEFORE you size.
If you can't state your win rate and average R,
you're gambling, not trading.
8. Run Monte Carlo quarterly.
Your actual stats change. Your sizing should adapt.
Related Skills
- Risk And Portfolio
- Drawdown Playbook
- Portfolio Optimization
- Real-Time Risk Monitor