Advanced Topics
This section covers sophisticated techniques for users who want to customize their analysis beyond the standard framework. Topics include custom loss distributions, correlation modeling, multi-year optimization, and advanced stochastic processes.
Customizing Loss Distributions
Industry-Specific Loss Modeling
Different industries have unique loss patterns. Here’s how to customize the distributions for your sector:
Custom loss distribution for cyber risks
from ergodic_insurance.loss_distributions import CustomLossDistribution
import numpy as np
from scipy import stats
class CyberLossDistribution(CustomLossDistribution):
"""Model cyber losses with increasing frequency over time."""
def __init__(self, base_frequency=0.5, growth_rate=0.15):
self.base_frequency = base_frequency
self.growth_rate = growth_rate # Cyber risk growing 15% annually
def generate_losses(self, year, random_state=None):
# Frequency increases over time
current_frequency = self.base_frequency * (1 + self.growth_rate) ** year
# Number of events (Poisson process)
n_events = random_state.poisson(current_frequency)
# Severity follows power law (heavy tail)
if n_events > 0:
# Power law: many small, few catastrophic
alpha = 2.5 # Shape parameter
x_min = 100_000 # Minimum loss
# Generate from power law
u = random_state.uniform(0, 1, n_events)
losses = x_min * (1 - u) ** (-1 / (alpha - 1))
return losses
return np.array([])
Fitting Distributions to Historical Data
Use your actual loss history to calibrate distributions:
Calibrating distributions from data
from ergodic_insurance.loss_distributions import fit_distribution
import pandas as pd
from scipy import stats
# Load your historical losses
loss_data = pd.read_csv('historical_losses.csv')
# Separate by magnitude
small_losses = loss_data[loss_data['amount'] < 100_000]['amount']
large_losses = loss_data[loss_data['amount'] >= 100_000]['amount']
# Fit frequency distribution
years_of_data = loss_data['year'].nunique()
small_frequency = len(small_losses) / years_of_data
large_frequency = len(large_losses) / years_of_data
# Fit severity distributions
# Test multiple distributions
distributions = [
stats.lognorm,
stats.gamma,
stats.weibull_min,
stats.pareto
]
best_fit = None
best_ks_stat = float('inf')
for dist in distributions:
# Fit distribution
params = dist.fit(large_losses)
# Kolmogorov-Smirnov test
ks_stat, p_value = stats.kstest(large_losses,
lambda x: dist.cdf(x, *params))
if ks_stat < best_ks_stat and p_value > 0.05:
best_fit = (dist, params)
best_ks_stat = ks_stat
print(f"Best fitting distribution: {best_fit[0].name}")
print(f"Parameters: {best_fit[1]}")
Correlation Modeling
Modeling Business Cycle Correlation
Losses often correlate with economic conditions:
Implementing correlation between revenue and losses
from ergodic_insurance.stochastic_processes import CorrelatedShocks
class CorrelatedRiskModel:
"""Model correlation between business performance and losses."""
def __init__(self, correlation_matrix):
"""
correlation_matrix: 2x2 matrix
[[1.0, rho],
[rho, 1.0]]
where rho is correlation between revenue shock and loss shock
"""
self.correlation_matrix = correlation_matrix
def generate_shocks(self, n_periods, random_state=None):
"""Generate correlated shocks for revenue and losses."""
# Generate independent standard normal variables
if random_state is None:
random_state = np.random.RandomState()
independent_shocks = random_state.randn(n_periods, 2)
# Apply Cholesky decomposition for correlation
L = np.linalg.cholesky(self.correlation_matrix)
correlated_shocks = independent_shocks @ L.T
return {
'revenue_shocks': correlated_shocks[:, 0],
'loss_shocks': correlated_shocks[:, 1]
}
# Example: 30% correlation between bad revenue and high losses
correlation_model = CorrelatedRiskModel(
correlation_matrix=[[1.0, 0.3],
[0.3, 1.0]]
)
# Use in simulation
shocks = correlation_model.generate_shocks(n_periods=10)
for year in range(10):
revenue_multiplier = 1 + 0.15 * shocks['revenue_shocks'][year]
loss_multiplier = np.exp(0.3 * shocks['loss_shocks'][year])
# Bad revenue years have higher losses
annual_revenue = base_revenue * revenue_multiplier
annual_losses = base_losses * loss_multiplier
Geographic and Peril Correlation
Model correlation across locations and perils:
Multi-location correlation modeling
class MultiLocationRisk:
"""Model correlated risks across multiple locations."""
def __init__(self, locations, correlation_by_distance):
self.locations = locations
self.correlation_func = correlation_by_distance
def build_correlation_matrix(self):
"""Build correlation matrix based on geographic distance."""
n = len(self.locations)
corr_matrix = np.eye(n)
for i in range(n):
for j in range(i+1, n):
# Calculate distance between locations
distance = self.calculate_distance(
self.locations[i],
self.locations[j]
)
# Correlation decreases with distance
correlation = self.correlation_func(distance)
corr_matrix[i, j] = correlation
corr_matrix[j, i] = correlation
return corr_matrix
def calculate_distance(self, loc1, loc2):
"""Calculate distance between two locations."""
# Simplified Euclidean distance
return np.sqrt((loc1['lat'] - loc2['lat'])**2 +
(loc1['lon'] - loc2['lon'])**2)
# Example: Correlation decreases exponentially with distance
locations = [
{'name': 'Factory A', 'lat': 40.7, 'lon': -74.0},
{'name': 'Factory B', 'lat': 41.8, 'lon': -87.6},
{'name': 'Factory C', 'lat': 34.0, 'lon': -118.2}
]
correlation_func = lambda d: np.exp(-d / 500) # 500 mile correlation length
multi_location = MultiLocationRisk(locations, correlation_func)
corr_matrix = multi_location.build_correlation_matrix()
Multi-Year Optimization
Dynamic Insurance Strategies
Optimize insurance purchases over multiple years:
Multi-year insurance optimization
class DynamicInsuranceOptimizer:
"""Optimize insurance strategy over multiple years."""
def __init__(self, planning_horizon=5):
self.planning_horizon = planning_horizon
def optimize_multi_year(self, manufacturer, market_conditions):
"""Find optimal insurance path over planning horizon."""
# State space: (assets, market_cycle, years_remaining)
states = self.discretize_state_space(manufacturer)
# Value function: V(state, action) = expected future value
value_function = {}
optimal_policy = {}
# Backward induction (dynamic programming)
for year in range(self.planning_horizon, 0, -1):
for state in states:
best_value = -np.inf
best_action = None
# Try different insurance levels
for retention in [100_000, 250_000, 500_000, 1_000_000]:
for limit in [5_000_000, 15_000_000, 25_000_000]:
# Calculate expected value of this action
action = (retention, limit)
expected_value = self.calculate_expected_value(
state, action, year, value_function
)
if expected_value > best_value:
best_value = expected_value
best_action = action
value_function[(state, year)] = best_value
optimal_policy[(state, year)] = best_action
return optimal_policy
def calculate_expected_value(self, state, action, year, value_function):
"""Calculate expected value of action in given state."""
retention, limit = action
assets, market_state = state
# Simulate outcomes
outcomes = []
for scenario in range(100): # Sample scenarios
# Apply insurance for this year
premium = self.calculate_premium(retention, limit, market_state)
# Generate losses
losses = self.generate_losses(market_state)
# Apply insurance
retained_loss = min(losses, retention)
insurance_recovery = min(max(losses - retention, 0), limit)
# Update assets
new_assets = assets * (1 + growth_rate) - premium - retained_loss
# Transition to new market state
new_market_state = self.market_transition(market_state)
# Future value (if not final year)
if year > 1:
future_value = value_function.get(
((new_assets, new_market_state), year - 1), 0
)
else:
future_value = new_assets
outcomes.append(future_value)
return np.mean(outcomes)
Market Cycle Timing
Adjust insurance purchases based on market cycles:
Market-aware insurance purchasing
class MarketCycleStrategy:
"""Adjust insurance based on market conditions."""
def __init__(self):
self.market_states = ['hard', 'transitioning', 'soft']
self.transition_matrix = [
[0.5, 0.4, 0.1], # From hard market
[0.3, 0.4, 0.3], # From transitioning
[0.1, 0.4, 0.5] # From soft market
]
def recommend_strategy(self, current_market, company_state):
"""Recommend insurance strategy based on market."""
if current_market == 'hard':
# Hard market: prices high, capacity limited
return {
'strategy': 'defensive',
'retention': 'increase by 25%',
'limit': 'maintain current',
'timing': 'lock in multi-year if possible',
'alternative': 'consider captive or self-insurance'
}
elif current_market == 'soft':
# Soft market: prices low, capacity abundant
return {
'strategy': 'aggressive',
'retention': 'decrease to optimal',
'limit': 'increase by 50%',
'timing': 'lock in long-term',
'alternative': 'buy additional coverage'
}
else: # transitioning
return {
'strategy': 'balanced',
'retention': 'maintain',
'limit': 'modest increase',
'timing': 'short-term contracts',
'alternative': 'prepare for market shift'
}
Advanced Stochastic Processes
Jump Diffusion Models
Model sudden shocks alongside continuous volatility:
Jump diffusion for catastrophic events
class JumpDiffusionProcess:
"""Combine continuous volatility with discrete jumps."""
def __init__(self, drift=0.06, volatility=0.15,
jump_intensity=0.1, jump_mean=-0.3, jump_std=0.2):
self.drift = drift
self.volatility = volatility
self.jump_intensity = jump_intensity # Jumps per year
self.jump_mean = jump_mean # Average jump size (negative = down)
self.jump_std = jump_std
def simulate(self, S0, T, dt, random_state=None):
"""Simulate jump diffusion process."""
if random_state is None:
random_state = np.random.RandomState()
n_steps = int(T / dt)
times = np.linspace(0, T, n_steps + 1)
S = np.zeros(n_steps + 1)
S[0] = S0
for i in range(1, n_steps + 1):
# Brownian motion component
dW = random_state.randn() * np.sqrt(dt)
# Jump component
dN = random_state.poisson(self.jump_intensity * dt)
if dN > 0:
# Jump occurred
jump_size = random_state.normal(
self.jump_mean,
self.jump_std,
dN
).sum()
else:
jump_size = 0
# Combine diffusion and jump
S[i] = S[i-1] * np.exp(
(self.drift - 0.5 * self.volatility**2) * dt +
self.volatility * dW +
jump_size
)
return times, S
# Example: Revenue with occasional major disruptions
revenue_process = JumpDiffusionProcess(
drift=0.06, # 6% growth
volatility=0.10, # 10% regular volatility
jump_intensity=0.2, # One jump every 5 years on average
jump_mean=-0.25, # 25% average revenue drop when jump occurs
jump_std=0.15 # Jump size volatility
)
Regime-Switching Models
Model different economic regimes:
Regime-switching volatility model
class RegimeSwitchingModel:
"""Switch between different volatility regimes."""
def __init__(self):
self.regimes = {
'stable': {'growth': 0.06, 'volatility': 0.08},
'volatile': {'growth': 0.04, 'volatility': 0.25},
'crisis': {'growth': -0.05, 'volatility': 0.40}
}
# Transition probabilities (monthly)
self.transition_matrix = {
'stable': {'stable': 0.95, 'volatile': 0.04, 'crisis': 0.01},
'volatile': {'stable': 0.10, 'volatile': 0.85, 'crisis': 0.05},
'crisis': {'stable': 0.05, 'volatile': 0.25, 'crisis': 0.70}
}
def simulate(self, initial_regime, n_periods):
"""Simulate regime switches and returns."""
current_regime = initial_regime
regimes = []
returns = []
for period in range(n_periods):
regimes.append(current_regime)
# Generate return for current regime
params = self.regimes[current_regime]
monthly_return = np.random.normal(
params['growth'] / 12,
params['volatility'] / np.sqrt(12)
)
returns.append(monthly_return)
# Transition to next regime
probs = self.transition_matrix[current_regime]
current_regime = np.random.choice(
list(probs.keys()),
p=list(probs.values())
)
return regimes, returns
Alternative Risk Transfer
Parametric Insurance Design
Design parametric triggers for automatic payouts:
Parametric insurance for business interruption
class ParametricInsurance:
"""Insurance that pays based on objective triggers."""
def __init__(self, trigger_type='revenue_drop'):
self.trigger_type = trigger_type
def design_revenue_trigger(self, baseline_revenue):
"""Design parametric trigger based on revenue drop."""
triggers = [
{
'level': 'minor',
'threshold': 0.80 * baseline_revenue, # 20% drop
'payout': 1_000_000,
'base_premium_rate': 0.02
},
{
'level': 'major',
'threshold': 0.60 * baseline_revenue, # 40% drop
'payout': 5_000_000,
'base_premium_rate': 0.01
},
{
'level': 'severe',
'threshold': 0.40 * baseline_revenue, # 60% drop
'payout': 15_000_000,
'base_premium_rate': 0.005
}
]
return triggers
def calculate_payout(self, actual_revenue, triggers):
"""Calculate parametric payout based on actual results."""
total_payout = 0
for trigger in triggers:
if actual_revenue < trigger['threshold']:
total_payout += trigger['payout']
return total_payout
Captive Insurance Optimization
Evaluate forming a captive insurance company:
Captive insurance feasibility analysis
class CaptiveAnalysis:
"""Analyze feasibility of captive insurance company."""
def __init__(self, parent_company):
self.parent = parent_company
def evaluate_captive(self, retention_level, premium_volume):
"""Evaluate captive insurance economics."""
# Initial capital requirements
initial_capital = max(
retention_level * 3, # 3x annual retention
5_000_000 # Regulatory minimum
)
# Operating costs
annual_costs = {
'management': 250_000,
'regulatory': 100_000,
'actuarial': 75_000,
'audit': 50_000,
'reinsurance': premium_volume * 0.60 # 60% ceded
}
# Tax benefits (simplified)
tax_deduction = premium_volume * self.parent.tax_rate
# Investment income on reserves
investment_income = initial_capital * 0.04 # 4% return
# Calculate NPV over 10 years
cash_flows = []
for year in range(10):
# Inflows
premium_received = premium_volume
tax_benefit = tax_deduction
investment_return = investment_income
# Outflows
operating_cost = sum(annual_costs.values())
expected_claims = retention_level * 0.7 # 70% loss ratio
net_cash_flow = (
premium_received +
tax_benefit +
investment_return -
operating_cost -
expected_claims
)
cash_flows.append(net_cash_flow)
# Calculate NPV
discount_rate = 0.08
npv = -initial_capital
for i, cf in enumerate(cash_flows):
npv += cf / (1 + discount_rate) ** (i + 1)
return {
'initial_capital': initial_capital,
'annual_benefit': np.mean(cash_flows),
'npv': npv,
'breakeven_year': self.find_breakeven(cash_flows, initial_capital),
'recommended': npv > 0
}
Performance Optimization
Caching and Parallel Processing
Speed up large-scale simulations:
Optimized simulation with caching
from functools import lru_cache
from multiprocessing import Pool
import pickle
import hashlib
class OptimizedMonteCarloEngine:
"""High-performance Monte Carlo engine."""
def __init__(self, n_cores=None):
self.n_cores = n_cores or mp.cpu_count()
self.cache_dir = 'simulation_cache'
def run_parallel(self, params, n_simulations):
"""Run simulations in parallel."""
# Check cache first
cache_key = self.generate_cache_key(params, n_simulations)
cached_result = self.load_from_cache(cache_key)
if cached_result is not None:
print("Loaded from cache")
return cached_result
# Split simulations across cores
chunk_size = n_simulations // self.n_cores
chunks = [(params, chunk_size, i)
for i in range(self.n_cores)]
# Run in parallel
with Pool(self.n_cores) as pool:
results = pool.map(self.run_chunk, chunks)
# Combine results
combined = self.combine_results(results)
# Cache results
self.save_to_cache(cache_key, combined)
return combined
@staticmethod
def run_chunk(args):
"""Run a chunk of simulations."""
params, n_sims, seed = args
# Set unique seed for this chunk
np.random.seed(seed)
# Run simulations
results = []
for _ in range(n_sims):
result = run_single_simulation(params)
results.append(result)
return results
def generate_cache_key(self, params, n_simulations):
"""Generate unique cache key for parameters."""
# Serialize parameters
param_str = pickle.dumps((params, n_simulations))
# Generate hash
return hashlib.md5(param_str).hexdigest()
Next Steps
These advanced topics provide powerful tools for customization:
Start simple - Use basic framework first
Identify gaps - Where does standard model fall short?
Add complexity gradually - One advanced feature at a time
Validate thoroughly - Backtest against historical data
Document assumptions - Critical for advanced models
For questions about these advanced topics, consult the Frequently Asked Questions or review the full API documentation.