反向房屋抵押貸款之證券化- 四元樹模型之應用

Abstract

承做反向房屋抵押貸款有許多的風險，包括有利率風險、房屋價值風險和死亡率風險，而當反向房屋抵押貸款的貸款餘額超過抵押房屋的價值時，則反向房屋抵押貸款的發行機構將會面臨了臨界風險。本文中的利率模型採用Black-Derman-Toy模型(BDT)來生成未來短期利率的機率分布；而房價模型方面則採用Cox-Ross-Rubinstein模型(CRR) ，死亡率模型為Lee-Carter模型。另外，本篇使用了三維度的四元樹模擬方法，觀察在短期利率模型與房屋價值模型相關的條件下，貸放機構將會面臨的預期損失。另外，對於承做反向房屋抵押貸款的貸放機構而言，最高可貸成數是由貸放機構未來預期損失的淨現值總合等於未來貸款保費的淨現值總合所求得。然而，當貸放機構未來所遭遇的實質損失大於預期損失時，貸放機構則將有未預期損失，因此為了移轉此非預期損失，我們設計了一個證券化的模型，希望藉由發行債券的方式，將此反向房屋抵押貸款發生在臨界點之後的臨界風險移轉給資本市場中的債券持有人。

When the outstanding balance exceeds the housing value before the loan is settled, the insurer suffers an exposure to crossover risk induced by three risk factors: interest rates, house prices and mortality rates. Under the consideration of housing price risk, interest rate risk and longevity risk, we provide a three-dimensional lattice method which simultaneously captures the evolution of housing price and short-term interest rate to numerically calculate the fair valuation of reverse mortgages. For a mortgage reverse insurer, the maximum level of reverse mortgage insurance is determined by setting the present value of total expected claim losses equal to the present value of the premium charges. However, when the actual loss is higher than the expected loss, the insurer will incur an unexpected loss. To offset the potential loss, we also design a crossover bond, the payoff structure of which is related to the actual losses and expected losses, to transfer the unexpected loss into the bond investors. Therefore, through the crossover bonds, the reverse mortgage insurers can transfer the crossover risk into the bondholders.