Portfolio Optimization in Reinsurance | Kristina Williams, CQF

Abstract: We present a comprehensive framework for optimizing reinsurance portfolios under tail-risk constraints using stochastic programming techniques. By incorporating catastrophe loss distributions, correlation structures, and capital allocation requirements, we demonstrate how reinsurers can construct efficient portfolios that balance return objectives with regulatory capital constraints and risk appetite.

1. Introduction

Reinsurance portfolio construction presents unique challenges compared to traditional financial portfolios. Unlike equity or fixed-income assets that exhibit relatively continuous return distributions, reinsurance treaties are exposed to catastrophic tail events with highly skewed loss distributions.

2. Mathematical Framework

2.1 Portfolio Loss Distribution

Consider a reinsurance portfolio consisting of $n$ treaties. Let $L_i$ denote the random loss from treaty $i$, and $w_i$ the proportion of capital allocated to treaty $i$. The aggregate portfolio loss is:

$$L_p = \sum_{i=1}^{n} w_i L_i$$

2.2 Optimization Problem

We formulate the portfolio optimization as a constrained stochastic program:

$$\begin{aligned} \max_{w} \quad & \mathbb{E}[R_p] - \lambda \cdot \text{CVaR}_\alpha(L_p) \\ \text{s.t.} \quad & \sum_{i=1}^{n} w_i = 1 \\ & \text{VaR}_{0.995}(L_p) \leq K_{cap} \end{aligned}$$

Figure 1: Efficient frontier showing the return versus CVaR₉₉.₅ tradeoff. The red point indicates the optimal portfolio under a capital constraint. Hover for exact values.

3. Results

3.1 Portfolio Composition

Applying our framework to a sample portfolio of property catastrophe treaties, we obtain the following optimal allocation:

Treaty	Peril	Region	Weight (%)	Expected Loss Ratio
US Wind QS	Hurricane	Gulf Coast	12.3	68%
CA Earthquake XL	Earthquake	California	8.5	45%
EU Windstorm QS	Windstorm	Western Europe	15.7	72%
JP Earthquake XL	Earthquake	Japan	9.2	38%
Other treaties	Various	Global	54.3	55%

Figure 2: Optimal portfolio weights by treaty.

Figure 3: Correlation matrix of treaty losses estimated from Monte Carlo scenarios. Note the strong correlation between US Wind and CA Earthquake.

Figure 4: Simulated aggregate loss distribution with VaR and CVaR markers. Zoom and pan to explore the tail.

4. Conclusion

This framework demonstrates how modern portfolio theory can be adapted for reinsurance applications by incorporating heavy-tailed loss distributions, complex dependence structures, and regulatory capital constraints.

References

Rockafellar, R.T., & Uryasev, S. (2000). "Optimization of conditional value-at-risk." Journal of Risk, 2(3), 21-42.
McNeil, A.J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management. Princeton University Press.

Portfolio Optimization in Reinsurance: A Stochastic Programming Approach