# An Integrated Approach to Pricing Catastrophe Reinsurance

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## Abstract

**:**

## 1. Introduction

**Layer premium = Expected layer loss + Risk load factor × Standard Deviation,**

- (1)
- We first model a default-risky reinsurer by employing Merton’s (1974) structural approach to endogenize default.
- –
- On the asset side, since the reinsurer holds a large proportion of fixed-income assets in the asset portfolio, we model the asset dynamics taking into account explicitly the impact of stochastic interest rates. We make a measure change from the physical pricing measure to the equivalent martingale pricing measure Q by embedding the market price of interest rate risk.
- –
- On the liability side, since the reinsurer’s non-catastrophic liability shocks are idiosyncratic and small, we apply the law of large numbers to assume away this risk premium. We also make a measure change from the physical pricing measure to the equivalent martingale pricing measure Q by embedding the market price of interest rate risk.

- (2)
- We model catastrophe arrivals. We make use of the empirical finding that catastrophe derivatives are zero-beta assets, and thus both the loss number and the amount of losses have zero risk premiums.
- (3)
- We proceed to price a default-risky reinsurance contract under the equivalent martingale pricing measure Q as a martingale.
- (4)
- Finally, we extend the pricing formula to incorporate risk load/markup to account for the observed empirical characteristics of the (re)insurance market. The interpretation of the markup is the same as in the traditional actuarial pricing approach except that (1) our expectation is taken with respect to the risk-neutralized martingale pricing measure, but in the traditional approach the expectation is taken with respect to the physical pricing measure; and (2) our markup only needs to account for market imperfections and other idiosyncratic factors, while the markup in the traditional actuarial approach accounts for both modeled and non-modeled factors.

## 2. Modeling Reinsurer Default with Asset, Interest Rate, Liability, and Catastrophe Loss

#### Dynamics

**Theorem**

**1.**

**Proof.**

_{L},

_{t}, pertain to idiosyncratic shocks to the capital market, we assume a zero risk premium for this risk.

**Lemma**

**1.**

**Proof.**

_{t}, is described as follows:

## 3. Pricing a Cat Reinsurance Contract and the Monte Carlo Simulation Results

## 4. Concluding Remarks

## Author Contributions

## Conflicts of Interest

## Appendix A. Procedures of the Monte Carlo Simulation Method

_{R,0}using Equation (15). Because the premium depends on the values of the reinsurer’s assets and liabilities, we have to specify the stochastic processes for these variables (Equations (9) and (11)). Applying Ito’s lemma to the logarithm of the value of a reinsurer’s assets, Equation (9) becomes the following system:

_{R,0}can be easily calculated via averaging over the contingent payoffs corresponding to the simulated values.

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1 | Traditional reinsurers tend to invest in illiquid and information-intensive financial activities, and so they charge premiums based on correlations with their own pre-existing portfolios and U.S. nationwide cat risks, rather than with any market portfolio, resulting in significantly higher cost of capital. |

2 | A martingale is a stochastic variable with no drift such that the current expectation of the future value of the variable is always equal to the current value of the variable. The above authors show that the absence of arbitrage is equivalent to the existence of an equivalent martingale pricing measure Q (or risk-neutral/risk-neutralized pricing measure as used by some authors) such that all normalized (with respect to a chosen numeraire) security prices are martingales and as such they can be priced by taking expectations under the measure Q. The beauty of martingale pricing is that it applies to both complete and incomplete markets as “absence of arbitrage” is the only required assumption. When markets are complete with the absence of arbitrage, martingale pricing theory guarantees that the equivalent martingale measure is unique, thus the market price of risk does not explicitly enter into the valuation process. In this context, options can be priced preference-free as if agents were risk-neutral as in the Black–Scholes model. When markets are incomplete, however, the absence of arbitrage no longer guarantees a unique martingale measure. In this case, information related to the market prices of risk that embeds the risk-aversion behavior of agents is needed to uniquely identify the equivalent martingale measure (see Geman (2005) for a review). The procedure of embedding the market price of risk to obtain such a unique measure is often called risk-neutralization. |

3 | Gürtler et al. (2012) show that this premium can be significant when a mega catastrophe strikes, but we assume this scenario away in this paper. |

4 | |

5 |

Asset Parameters | ||

$V$ | Reinsurer’s assets | V/L = 1.3 |

${\mu}_{V}$ | Drift due to credit risk | Irreverent |

${\varphi}_{V}$ | Interest rate elasticity of asset | −3 |

${\sigma}_{v}$ | Volatility of credit risk | 5% |

${W}_{V,t}$ | Wiener process for credit shocks | |

Liability Parameters | ||

$L$ | Reinsurer’s liabilities | 100 |

${\mu}_{L}$ | Drift due to idiosyncratic risk | 0 |

${\varphi}_{L}$ | Interest rate elasticity of liability | −3 |

${\sigma}_{L}$ | Volatility of idiosyncratic risk | 2% |

${W}_{L,t}$ | Wiener process for idiosyncratic shocks | |

Interest Rate Parameters | ||

r | Initial instantaneous interest rate | 2% |

κ | Magnitude of mean-reverting force | 0.2 |

$m$ | Long-run mean of interest rate | 5% |

$\nu $ | Volatility of interest rate | 10% |

${\lambda}_{r}$ | Market price of interest rate risk | −0.01 |

$Z$ | Wiener process for interest rate shocks | |

Catastrophe loss Parameters for C_{t} | ||

$N(t)$ | Poisson process for the arrival of catastrophes | |

$\lambda $ | Catastrophe arrival intensity | 0.5 |

${\mu}_{C}$ | Mean of the logarithm of the losses per arrival | 2 |

${\sigma}_{C}$ | Standard deviation of the logarithm of the losses per arrival | 0.5 |

Other Parameters | ||

A | Attachment level of a reinsurance contract | 10~30 |

$M$ | Cap level of loss paid by a reinsurance contract | 60~90 |

T | Maturity | 3 years |

u | Reinsurance markup | 0.4 |

Coverage | M = 60 | M = 65 | M = 70 | M = 75 | M = 80 | M = 85 | M = 90 |
---|---|---|---|---|---|---|---|

A = 10 | 7.26957 | 7.28648 | 7.29306 | 7.30003 | 7.30261 | 7.30284 | 7.30634 |

A = 15 | 4.45191 | 4.46950 | 4.47824 | 4.48331 | 4.48579 | 4.48792 | 4.99274 |

A = 20 | 2.73468 | 2.75143 | 2.76143 | 2.76739 | 2.16897 | 2.17120 | 2.17444 |

A = 25 | 1.57234 | 1.59048 | 1.59951 | 1.60448 | 1.60706 | 1.60929 | 1.61162 |

A = 30 | 0.93787 | 0.98590 | 1.05583 | 1.06080 | 1.06338 | 1.06561 | 1.06740 |

(λ, ${\mathit{\sigma}}_{\mathit{C}}$) | Reinsurance Price $(1+\mathit{u})\mathit{P}{\mathit{V}}_{\mathit{R},0}$ for Coverage Payer (90, 10) | ||
---|---|---|---|

V/L = 1.1 | V/L = 1.3 | V/L = 1.5 | |

(0.5, 0.5) | 7.30634 | 7.40443 | 7.58688 |

(1, 0.5) | 13.24426 | 13.78995 | 14.12347 |

(2, 0.5) | 19.56788 | 19.98765 | 20.31452 |

(0.5, 1) | 20.75633 | 21.24536 | 21.76542 |

(1, 1) | 24.18776 | 24.35473 | 24.87682 |

(2, 1) | 27.78653 | 28.89672 | 29.45328 |

(0.5, 2) | 40.28763 | 41.69782 | 43.13476 |

(1, 2) | 43.21675 | 43.87862 | 44.34724 |

(2, 2) | 48.8976 | 49.90163 | 51.27658 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chang, C.W.; Chang, J.S.K. An Integrated Approach to Pricing Catastrophe Reinsurance. *Risks* **2017**, *5*, 51.
https://doi.org/10.3390/risks5030051

**AMA Style**

Chang CW, Chang JSK. An Integrated Approach to Pricing Catastrophe Reinsurance. *Risks*. 2017; 5(3):51.
https://doi.org/10.3390/risks5030051

**Chicago/Turabian Style**

Chang, Carolyn W., and Jack S. K. Chang. 2017. "An Integrated Approach to Pricing Catastrophe Reinsurance" *Risks* 5, no. 3: 51.
https://doi.org/10.3390/risks5030051