Frequently Asked Questions

General Questions

What is ergodic theory and why does it matter for insurance?

Answer: Ergodic theory studies the long-term average behavior of systems. For insurance, it reveals a critical distinction:

Ensemble average: What happens on average across many companies (traditional approach)
Time average: What YOUR specific company experiences over time (ergodic approach)

For multiplicative processes like business growth, these averages diverge. A company that goes bankrupt can’t benefit from the “average” success of others. Ergodic optimization ensures YOUR company survives and thrives, not just the statistical average.

How is this different from traditional actuarial analysis?

Answer: Traditional actuarial analysis focuses on expected values and fair premiums. Our approach recognizes that:

You can’t diversify across time - A bad year affects all future years
Ruin is permanent - Bankruptcy means game over, not a temporary setback
Growth compounds - Missing growth in one year affects all future wealth
Time-average return is what you actually experience - Not the ensemble average

The result: Insurance that seems “expensive” by traditional metrics can be optimal for long-term growth.

What’s the minimum company size to benefit from this analysis?

Answer: Any company with:

At least $1M in assets
Annual revenues > $500K
Exposure to losses > 5% of assets

Can benefit from ergodic optimization. Smaller companies often benefit MORE because they have less ability to absorb large losses.

Technical Questions

How many simulations do I need to run?

Answer: It depends on your purpose:

Purpose                  Minimum    Recommended   Optimal
─────────────────────────────────────────────────────────
Initial exploration      1,000      5,000         10,000
Detailed analysis        5,000      10,000        50,000
Final decisions         10,000      50,000       100,000
Board presentation      50,000     100,000       500,000

Check convergence using:

# Test if results have stabilized
from ergodic_insurance.convergence import test_convergence

stable = test_convergence(results, window=1000, tolerance=0.001)
if not stable:
    print("Increase simulation count")

Why do my results vary between runs?

Answer: Monte Carlo simulation is probabilistic. Variation is normal but should decrease with more simulations:

Standard error decreases as 1/√n
Use random seeds for reproducibility
Consider confidence intervals, not just point estimates

# For reproducible results
engine = MonteCarloEngine(n_simulations=10000, random_seed=42)

# Calculate confidence intervals
lower_95 = np.percentile(results, 2.5)
upper_95 = np.percentile(results, 97.5)
print(f"95% CI: [{lower_95:.2f}, {upper_95:.2f}]")

How do I model correlation between risks?

Answer: Use the correlation features in the advanced topics:

# Simple correlation between revenue and losses
correlation = 0.3  # 30% correlation

# In bad revenue years, losses are higher
if revenue_shock < 0:
    loss_multiplier = 1 + correlation * abs(revenue_shock)
    adjusted_losses = base_losses * loss_multiplier

For complex correlations, see Advanced Topics.

What if I don’t have good historical loss data?

Answer: Use industry benchmarks and conservative assumptions:

Start with industry data - Insurance brokers can provide
Use conservative parameters - Better to overestimate risk initially
Sensitivity analysis - Test how results change with different assumptions
Update regularly - Refine as you gather data

# Conservative approach without data
conservative_losses = {
    'frequency': industry_average * 1.5,  # 50% higher
    'severity_mean': industry_severity * 1.2,  # 20% higher
    'severity_cv': 1.5  # High uncertainty
}

Implementation Questions

How do I convince management to increase insurance spending?

Answer: Focus on value creation, not cost:

Present as investment: Show ROI in terms of growth rate improvement
Quantify survival improvement: “Reduces 10-year bankruptcy risk from 15% to 2%”
Show peer comparisons: What similar successful companies do
Calculate opportunity cost: Value destroyed by inadequate coverage
Use visualizations: Graphs showing wealth paths with/without insurance

Example presentation points:

Current State:
- Premium: $200K/year (2% of assets)
- 10-year survival: 75%
- Expected growth: 5%/year
- 10-year expected value: $12M

Optimized Structure:
- Premium: $500K/year (5% of assets)
- 10-year survival: 95%
- Expected growth: 7.5%/year
- 10-year expected value: $18M

Net Benefit: +$6M (12x the additional premium!)

What if insurers won’t provide the coverage I need?

Answer: Several strategies:

Layer with multiple carriers - Split large limits across insurers
Alternative markets - Bermuda, Lloyd’s, captives
Parametric products - Objective triggers, easier to place
Gradual increase - Build relationships and track record
Risk mitigation - Demonstrate improvements to get better terms

How often should I review my insurance structure?

Answer:

Mandatory reviews: * Annually before renewal * After any loss > retention * When assets change > 25% * After M&A activity

Recommended reviews: * Quarterly dashboard check * Semi-annual optimization run * After significant market changes

Common Pitfalls

“Our losses are predictable, so we don’t need much insurance”

Answer: This is the “turkey problem” - a turkey might think life is predictable until Thanksgiving.

Past stability doesn’t guarantee future stability
Black swan events happen
Correlation emerges in crisis (everything goes wrong at once)
The cost of being wrong is bankruptcy

Always model tail risks, even if they haven’t happened yet.

“We’re too small for sophisticated insurance”

Answer: Actually, smaller companies need MORE sophisticated insurance because:

Less diversification = higher risk concentration
Limited access to emergency capital
One bad event can end the business
Growth depends on consistent reinvestment

The framework works especially well for companies under $50M in assets.

“Insurance is too expensive in hard markets”

Answer: Hard markets (high prices) are when insurance is MOST valuable:

Higher prices = higher perceived risk
Your competitors are also struggling with costs
Surviving hard markets creates competitive advantage
Lock in coverage before it becomes unavailable

Consider alternative structures in hard markets but don’t go naked.

“We can self-insure with reserves”

Answer: Self-insurance has hidden costs:

Opportunity cost: Reserves can’t be invested in growth
Concentration risk: One large loss depletes reserves
Credit impact: Reserves don’t provide same comfort to lenders
Tax inefficiency: No deduction for reserved amounts

True self-insurance requires 3-5x the insurance limit in liquid reserves.

Modeling Questions

How do I model a new risk we haven’t experienced?

Answer: Use scenario analysis and industry data:

# Model emerging cyber risk
cyber_scenarios = [
    {'probability': 0.60, 'impact': 500_000},    # Minor breach
    {'probability': 0.30, 'impact': 2_000_000},  # Significant breach
    {'probability': 0.09, 'impact': 10_000_000}, # Major breach
    {'probability': 0.01, 'impact': 50_000_000}  # Catastrophic
]

# Convert to frequency/severity
frequency = sum(s['probability'] for s in cyber_scenarios)
severity_mean = sum(s['probability'] * s['impact'] for s in cyber_scenarios) / frequency

What growth rate should I use?

Answer: Use your historical average but adjust for:

Business cycle stage
Industry maturity
Competitive dynamics
Investment plans

# Weighted average approach
historical_growth = 0.08  # Past 5 years
industry_growth = 0.05    # Industry average
plan_growth = 0.10        # Business plan

# Weighted by confidence
expected_growth = (
    0.5 * historical_growth +
    0.3 * industry_growth +
    0.2 * plan_growth
)

How do I account for inflation?

Answer: Model in real or nominal terms consistently:

Real terms (recommended): * Adjust all values for inflation * Use real growth rates * Easier to interpret

Nominal terms: * Include inflation in growth rates * Adjust severity trends * More complex but sometimes required

# Inflation adjustment
inflation_rate = 0.03  # 3% annual

# Real to nominal conversion
nominal_growth = real_growth + inflation_rate
nominal_losses = real_losses * (1 + inflation_rate) ** year

Troubleshooting

“The simulation is taking too long”

Answer: Several optimization strategies:

Reduce simulations for exploration (increase for final decisions)
Use parallel processing - See advanced topics
Cache results - Don’t re-run unchanged scenarios
Profile code - Find bottlenecks

# Enable caching
from functools import lru_cache

@lru_cache(maxsize=128)
def expensive_calculation(params):
    # Cached computation
    return result

“I’m getting negative wealth values”

Answer: This indicates ruin scenarios. Check:

Is your retention too high?
Are losses properly capped by insurance limits?
Is the model allowing recovery from negative wealth?

# Prevent negative wealth (ruin is absorbing)
if wealth <= 0:
    wealth = 0
    is_ruined = True
    # No further simulation for this path

“Results seem too good/bad to be true”

Answer: Common calibration issues:

Check units - Millions vs thousands
Verify rates - Annual vs monthly
Loss frequencies - Per year, not per simulation
Premium calculations - Percentage of limit, not assets
Time horizons - Consistent across comparisons

Still Have Questions?

If your question isn’t answered here:

Check the Glossary for term definitions
Review Advanced Topics for complex scenarios
Examine the example notebooks in ergodic_insurance/notebooks/
Consult the API documentation
Contact support with specific details

Remember: No question is too basic. Better to ask than to make costly mistakes in your insurance decisions.