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Cost-of-capital valuation with risky assets

Hansjörg Albrecher, Filip Lindskog, Hervé Zumbach

Abstract

Cost-of-capital valuation is a well-established approach to the valuation of liabilities and is one of the cornerstones of current regulatory frameworks for the insurance industry. Standard cost-of-capital considerations typically rely on the assumption that the required buffer capital is held in risk-less one-year bonds. The aim of this work is to analyze the effects of allowing investments of the buffer capital in risky assets, e.g.~in a combination of stocks and bonds. In particular, we make precise how the decomposition of the buffer capital into contributions from policyholders and investors varies as the degree of riskiness of the investment increases, and highlight the role of limited liability in the case of heavy-tailed insurance risks. We present a combination of general theoretical results, explicit results for certain stochastic models and numerical results that emphasize the key findings.

Cost-of-capital valuation with risky assets

Abstract

Cost-of-capital valuation is a well-established approach to the valuation of liabilities and is one of the cornerstones of current regulatory frameworks for the insurance industry. Standard cost-of-capital considerations typically rely on the assumption that the required buffer capital is held in risk-less one-year bonds. The aim of this work is to analyze the effects of allowing investments of the buffer capital in risky assets, e.g.~in a combination of stocks and bonds. In particular, we make precise how the decomposition of the buffer capital into contributions from policyholders and investors varies as the degree of riskiness of the investment increases, and highlight the role of limited liability in the case of heavy-tailed insurance risks. We present a combination of general theoretical results, explicit results for certain stochastic models and numerical results that emphasize the key findings.

Paper Structure

This paper contains 12 sections, 15 theorems, 93 equations, 13 figures.

Table of Contents

  1. Introduction
  2. A cost-of-capital valuation with risky assets
  3. Monotonicity properties
  4. The Gaussian model
  5. Value-at-Risk
  6. Expected Shortfall
  7. Numerical illustrations
  8. The Gaussian model
  9. The Lognormal model
  10. Lognormal investment returns and Pareto claims
  11. Conclusions
  12. Proofs

Key Result

Proposition 2.1

If $\rho$ is either $\operatorname{VaR}_{\alpha}$ or $\operatorname{ES}_{\alpha}$, $X_1$ and $S_1$ are independent and take nonnegative values only, $S_1$ is absolutely continuous and $\mathbb{P}(X_1=0) < 1-\alpha$, then there exists a unique $R_0>0$ solving r0.

Figures (13)

  • Figure 1: $R_{0}^{w}, C_{0}^{w}$ and $V_{0}^{w}$ for the Gaussian model with $\mu = 1.05, \gamma = 1,\nu =0.3$, $\rho=\operatorname{VaR}_{0.005}$ and various values of $\sigma$
  • Figure 2: $R_{0}^{w}, C_{0}^{w}$ and $V_{0}^{w}$ for the Gaussian model with $\mu = 1.05, \sigma=0.2,\gamma = 1$, $\rho=\operatorname{VaR}_{0.005}$ and various values of $\nu$
  • Figure 3: $R_{0}^{w}, C_{0}^{w}$ and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.05$, $\mathbb{E}[X_1] = 1$, $\text{std}(X)=0.3$, $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(S_1)$
  • Figure 4: Upper and lower bounds for $V_{0}^{w}$ in Figure \ref{['fig3']}
  • Figure 5: $R_{0}^{w}, C_{0}^{w}$ and $V_{0}^{w}$ for the lognormal model with $\mathbb{E}[S_1] = 1.05$, $\mathbb{E}[X_1] = 1$, $\text{std}(S_1)=0.2$, $\rho=\operatorname{VaR}_{0.005}$ and various values of $\text{std}(X_1)$
  • ...and 8 more figures

Theorems & Definitions (37)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.1
  • Proposition 2.1
  • Remark 2.4
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 27 more