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On a Class of Optimal Reinsurance Problems

N. D. Shyamalkumar, Tianrun Wang

Abstract

De Finetti's optimal reinsurance is a set of contracts, one for each risk in a portfolio, that caps the retained aggregate variance to a pre-specified level while minimizing total expected loss. The premiums are determined using the expected value principle, and the safety loading is allowed to vary with the risks. The original formulation assumed that the risks were independent and restricted contracts to quota shares on individual risks. A recent variation surprisingly yields a closed form for the contracts, while allowing dependence between risks and permitting the contracts to depend on all risks, without restricting their functional form. We extend this to the case of an arbitrary convex functional as the risk measure and use duality tools from convex analysis to show the equivalence between the constrained and the penalized versions of the underlying optimization problem. To explicitly solve the penalized version for the variance and the conditional value at risk (CVaR) as the risk measure, we resort to either variational analysis or a rudimentary approach. We show that a rudimentary approach can also address the choice of VaR, a non-convex functional, as the risk measure.

On a Class of Optimal Reinsurance Problems

Abstract

De Finetti's optimal reinsurance is a set of contracts, one for each risk in a portfolio, that caps the retained aggregate variance to a pre-specified level while minimizing total expected loss. The premiums are determined using the expected value principle, and the safety loading is allowed to vary with the risks. The original formulation assumed that the risks were independent and restricted contracts to quota shares on individual risks. A recent variation surprisingly yields a closed form for the contracts, while allowing dependence between risks and permitting the contracts to depend on all risks, without restricting their functional form. We extend this to the case of an arbitrary convex functional as the risk measure and use duality tools from convex analysis to show the equivalence between the constrained and the penalized versions of the underlying optimization problem. To explicitly solve the penalized version for the variance and the conditional value at risk (CVaR) as the risk measure, we resort to either variational analysis or a rudimentary approach. We show that a rudimentary approach can also address the choice of VaR, a non-convex functional, as the risk measure.
Paper Structure (6 sections, 7 theorems, 72 equations, 5 figures)

This paper contains 6 sections, 7 theorems, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Optimal Reinsurance Problem - Two Versions
  3. Variance as the Risk Measure
  4. Conditional Value at Risk as the Risk Measure
  5. Conclusion
  6. Appendix

Key Result

Theorem 1

For $\mathbf{X}\in L^1((\Omega,{\mathcal{F}},\mathbb{P}))^{\times n}$, let ${\mathcal{E}}\subseteq L^1(\Omega,{\mathcal{F}},\mathbb{P})$ be defined by Also, let the risk functional $\rho:{\mathcal{E}} \to \mathbb{R} \cup \{\pm \infty\}$ be convex, and weakly lower semi-continuous. Then, we have the following:

Figures (5)

  • Figure 1: Quotient Diagram
  • Figure 2: The left picture shows $h(\eta):=\eta-\mathbb{E}{\left(Z_{\eta}\right)}$ is increasing for an absolutely continuous $\mu$, with a zero at $\sigma=1.8029$. The right picture shows the objective of \ref{['Reins-P']} minimizes over all $R_i=\left(\sum_{i=k}^nX_i-\frac{\beta_k}{2\lambda^*}-\sigma\right)_+ \wedge X_k$ at this zero.
  • Figure 3: Fixed point iterations converge to the fixed point $\sigma=1.8029$. Initial point at $\eta_0=0.1$ followed by two iterations.
  • Figure 4: The plot shows the zero of $J_+'(q)$ corresponding to $\lambda^*=1.06\%$ under $c=5$
  • Figure 5: Left plot: $\lambda^*$ corresponding to the constraint ${\rm CVaR}_{\alpha}\left(Z^*\right)=c$ on the optimal retained risk. The red dashed lines correspond to the $c=5$ case. We notice that discontinuities at $0.01$ and $0.025$ corresponding to ${\beta_1}/{(1-\alpha)}$ and $\beta_2/(1-\alpha)$, respectively. Right plot: Solution of \ref{['con:q']}, $q^*$, is a continuous function of $\lambda$.

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • Example 1
  • Remark 2
  • Lemma 1
  • ...and 11 more