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Capital-Allocation-Induced Risk Sharing

Wing Fung Chong, Runhuan Feng, Kenneth Tsz Hin Ng

Abstract

This article proposes a new class of risk-sharing rules by exploring the relationship between capital allocation and risk sharing. While the former is concerned with ex-ante allocating capitals to different lines of business within a corporation based on the relationship among the individual risks, often also through the aggregate risk, the latter is an arrangement which collects risks from and allocates them to, also ex-ante, a group of participants. Drawing on this analogy, we introduce a novel idea of inducing risk-sharing rules by randomizing existing capital allocation principles. Such an approach derives new risk-sharing rules complementing known results in the literature, which were largely based on economic principles and Pareto optimality.

Capital-Allocation-Induced Risk Sharing

Abstract

This article proposes a new class of risk-sharing rules by exploring the relationship between capital allocation and risk sharing. While the former is concerned with ex-ante allocating capitals to different lines of business within a corporation based on the relationship among the individual risks, often also through the aggregate risk, the latter is an arrangement which collects risks from and allocates them to, also ex-ante, a group of participants. Drawing on this analogy, we introduce a novel idea of inducing risk-sharing rules by randomizing existing capital allocation principles. Such an approach derives new risk-sharing rules complementing known results in the literature, which were largely based on economic principles and Pareto optimality.

Paper Structure

This paper contains 38 sections, 8 theorems, 106 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Capital Allocation
  3. Risk Sharing
  4. Contributions and Organizations
  5. Related Literature
  6. Notation
  7. Randomization of Capital Allocation Principles
  8. General Approach
  9. Parameter Sampling Rule
  10. Inheritance of Properties
  11. Aggregate Capital Function
  12. Randomization of Top-Down Principles
  13. Optimization Principle
  14. Squared penalty
  15. Absolute penalty
  16. ...and 23 more sections

Key Result

Proposition 3.1

Let ${\bf K}:I\to\mathbb{R}^n$ be the family of the parametrized capital allocation principles in $\theta\in I$, with $K(\theta)=\sum_{i=1}^n K_i(\theta)$. The following statements hold.

Figures (7)

  • Figure 1: Illustration of capital-allocation-induced risk sharing rules
  • Figure 2: Implementation of induced risk-sharing rules based on top-down and bottom-up principles
  • Figure 3: Conditional expectations of individual risks given the aggregate risk
  • Figure 4: The realized sampling level $\Theta^{(j)}$ with $S^{(j)}=s$, $j=1,2$,
  • Figure 5: The induced risk sharing rules ${\bf H}({\bf X}^{(j)})$ with $S^{(j)}=s$, $j=1,2$.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Example 2.1: Marginal VaR, dhaene:2021
  • Definition 2.1
  • Proposition 3.1
  • proof
  • Example 3.1
  • Example 3.2
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Remark 4.1
  • ...and 31 more