Allianz Quantum Computing for Insurance Premium Pricing and Risk Aggregation

Allianz, the world’s largest insurer by assets, faces a computationally intensive problem at the heart of its underwriting business: aggregating correlated catastrophe losses across a diversified global portfolio. When a major hurricane or earthquake strikes, losses from flood, windstorm, and earthquake policies are not independent. Classical catastrophe risk platforms like RMS and AIR Worldwide address this through full Monte Carlo simulation, running 500,000 or more stochastic event scenarios to estimate the probability distribution of total portfolio loss. This distribution drives reinsurance purchasing, capital allocation under Solvency II, and premium pricing. The computational cost is substantial, and the quadratic convergence of Monte Carlo means that halving the estimation error requires four times as many scenarios. Allianz partnered with IBM Quantum to explore whether quantum amplitude estimation could provide the quadratic speedup that theory predicts.

The quantum approach encodes the correlated loss distribution as a quantum state. Allianz modeled inter-peril correlations using a Gaussian copula, which is standard practice in catastrophe modeling and captures the tail dependence structure between perils. The copula correlation matrix is decomposed via Cholesky factorization and encoded into a quantum circuit using a series of controlled rotations, producing a superposition over correlated loss scenarios. The marginal loss distributions for each peril (flood, earthquake, windstorm) are encoded using probability loading circuits derived from historical loss curves. The combined state represents the joint distribution over all perils and regions in the portfolio simultaneously. Conditional Value at Risk (CVaR) at the 99th percentile, a key regulatory metric under Solvency II internal model requirements, is extracted as the expectation value of a thresholded loss observable.

from qiskit_finance.circuit.library import LogNormalDistribution
from qiskit.circuit.library import LinearAmplitudeFunction
from qiskit_finance.applications.estimation import IterativeAmplitudeEstimation
import numpy as np

# Encode marginal loss distribution for a single peril (flood)
num_qubits = 5
mu_log = 12.5          # log-mean of loss (in millions)
sigma_log = 1.8        # log-std of loss
low, high = 0, 5000    # loss range in millions

flood_dist = LogNormalDistribution(
    num_target_qubits=num_qubits,
    mu=mu_log,
    sigma=sigma_log,
    bounds=(low, high)
)

# Define CVaR threshold: losses exceeding VaR_99
var_99 = 1200.0  # estimated 99th percentile loss (millions)
cvar_observable = LinearAmplitudeFunction(
    num_state_qubits=num_qubits,
    slope=1.0 / (high - low),
    offset=-var_99 / (high - low),
    domain=(low, high),
    image=(0, 1),
    breakpoints=[var_99]
)

# IQAE for CVaR estimation
iae = IterativeAmplitudeEstimation(
    epsilon_target=0.01,   # 1% absolute error target
    alpha=0.05,            # 95% confidence
)
result = iae.estimate(cvar_observable, flood_dist)
cvar_estimate = result.estimation * (high - low) + var_99
print(f"CVaR_99 estimate: ${cvar_estimate:.1f}M using {result.num_oracle_queries} circuit evaluations")

Allianz compared IQAE results against its production RMS RiskLink platform, which generates 500,000 event year scenarios to produce the exceedance probability curve. The IQAE implementation, running on IBM AerSimulator with noise models calibrated to Eagle 127Q characteristics, matched the classical CVaR estimate within 2% using only 2,048 circuit evaluations. The quadratic speedup manifests as a 244x reduction in the number of loss samples required to reach equivalent accuracy. On real Eagle 127Q hardware, decoherence and gate error limited circuit depth to around 15 layers, restricting the accuracy achievable without error mitigation. Allianz’s quantum roadmap identifies fault-tolerant hardware as the threshold for production deployment, with the current NISQ work establishing the algorithmic framework and validating the copula encoding methodology ahead of that transition.