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Optimal Insurance to Maximize Exponential Utility when Premium is Computed by a Convex Functional

Jingyi Cao, Dongchen Li, Virginia R. Young, Bin Zou

Abstract

We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.

Optimal Insurance to Maximize Exponential Utility when Premium is Computed by a Convex Functional

Abstract

We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity; then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.
Paper Structure (11 sections, 4 theorems, 43 equations, 3 figures, 1 algorithm)

This paper contains 11 sections, 4 theorems, 43 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimal insurance problem and its solution
  3. Statement of the problem
  4. Necessary condition for the optimal indemnity
  5. Algorithm for computing the optimal indemnity
  6. Existence of an optimal indemnity
  7. Examples
  8. Example 1
  9. Example 2
  10. Example 3
  11. Concluding remarks

Key Result

Theorem 2.1

Let $g \in \mathcal{G}$, and suppose there exists an optimal indemnity $\widehat{I}_g \in \mathcal{I}$ for Problem eq:insured_prob. Then, the optimal indemnity $\widehat{I}_g$ is given by in which the threshold $d_g$ is defined by with $M$ given by and, for every $x > 0$, the function $\kappa$ is defined by

Figures (3)

  • Figure 1: Convergence of $\{M_n\}$ defined by \ref{['eq:M_n']} to $M=3$ (left panel) and optimal indemnity $\hat{I}_g$ (right panel) for Example 1, in which $X \sim Exp(1)$, $g(x) = (4/3)x$, and $\gamma = 2$.
  • Figure 2: Convergence of $\{M_n\}$ defined by \ref{['eq:M_n']} to $M=5.4214$ (left panel) and optimal indemnity $\hat{I}_g$ (right panel) for Example 2, in which $X \sim Exp(1)$, $g(x) = x + x^2/2$, and $\gamma = 2$.
  • Figure 3: Convergence of $\{M_n\}$ defined by \ref{['eq:M_n']} to $M = 1.2288$ (left panel) and optimal indemnity $\hat{I}_g$ (right panel) for Example 3, in which $X \sim Exp(1)$, $g(x) = x + 0.1(x - 1)_+ + 0.2(x - 2)_+$, and $\gamma = 0.5$.

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Corollary 2.1
  • proof
  • Remark 2.2
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.3
  • ...and 1 more