死亡率巨災債券定價－－Transformed Gamma Distribution之運用

Pricing Mortality-Linked Securities with Transformed Gamma Distribution

壽命連結商品是保險公司規避人們長壽風險或死亡率巨災風險的工具，這些商品利用證券化的手法將可能的損失移轉給資本市場上的投資人。瑞士再保險公司於2003年底推出的Swiss Re mortality bond大受市場歡迎，本文以此商品為例探討壽命連結商品之訂價問題。文章採用均衡定價模型，假設死亡率指數服從Transformed Gamma分配，且假設投資人的期末財富、邊際效用及標的資產期末價值有特定形式下，得到一個風險資產評價關係式。此評價關係為風險中立，且壽命連結商品的價格即為其未來報酬以無風險利率折現。最終我們可推導出在Transformed Gamma分配下壽命連結商品的價格的封閉解，並以此封閉解來計算Swiss Re mortality bond的發行價格作為範例。

Due to frequent catastrophes all over the world in recent years, insurers face a larger mortality risk than before. To reduce this risk, the insurers have innovated mortality-linked securities to transfer systematic mortality risks to capital markets. In this article, assuming the mortality rate following a transformed gamma distribution, we derive an equilibrium valuation method to price the securities in a discrete time economy. The risk neutral valuation relationship is obtained, and the price of mortality-linked security is equal to the sum of expected payoffs discounted at a risk-free rate. Under the specific settings of investors’ preference, mortality rate, and wealth, the closed-form solutions for the mortality-linked security are obtained. Finally, we price the Swiss Re mortality bond issued in 2003 as a numerical example.