反向抵押貸款定價: 使用GARCH模型和NIG分配

Abstract

全球人口已經邁入高齡化社會，老年人口比例逐年攀升之已經是不可擋的趨勢，不少退休的老人無子女奉養，身邊只有房子而無錢過生活。因此在英美等國市場上近幾年發展出一種為老年人退休生活融資的工具─反向抵押貸款(Reverse Mortgage)。反向抵押貸款是指房屋所有權人達到一定年齡以後，將房屋抵押給銀行或保險公司而仍保有房屋之所有權，再由銀行或保險公司支付一定金額的養老金給申請人，直到申請人去世，其抵押的房屋就歸銀行或保險公司所有，而在貸款期間，並不需償還任何金錢予貸款人，也就是一種「以房養老」的觀念。反向抵押貸款的貸款價值取決於借款人的年齡，預期貸款利率，和房屋的價值。房屋價值是影響借貸總金額的最大因素，我們所使用的美國房價資料顯示房價的報酬率並非常態且有波動聚集的現象。在本文中，我們將利用GARCH模型和Normal Inverse Gaussian分配建構房價的隨機過程，並經由Duan(1995)所提出的局部風險中立評價法(LRNVR)和Conditional Esscher Transform轉換到風險中立測度的NIG-GARCH模型，最後再利用Lee-Carter(1992)隨機死亡率模型計算反向抵押貸款之公平保費和最大可貸乘數。

Today Life expectancy is getting longer globally. How to fund the retirement asset has become an important issue for the individual. To solve the problem of cash poor and equity rich for older persons, reverse mortgage has been introduced in developing countries such the U.S., the U.K. and Australia. Reverse mortgage is a new financial product that allows retirees to convert a proportion of the equity in their home into cash until they die. The loan value is determined by the borrower's age, the interest rate, and the home's value. The earlier models also assume house price returns follow normal distribution. Unfortunately, house price returns in our study are potentially non-normal. In this paper, we want to construct the house price model via GARCH with Normal Inverse Gaussian distribution (NIG-GARCH) option pricing model via local risk-neutral valuation relationship (LRNVR), conditional Esscher transform and mortality rates follows Lee-Carter (1992) model.