在Hull-White隨機利率及隨機死亡率下評價反向房屋抵押貸款

Abstract

各國的經濟正在負擔越來越多老年人口的問題，政府希望能解決老年人口所需要的醫療照護與收入，而發展出RM(Reverse Mortgage)，此商品將屋主的房屋權益抵押給銀行並轉換成收入，減輕扶養老年人口對社會造成的負擔。此篇在利率期限結構服從Hull-White模型、死亡率服從Lee-Carter模型，與房價服從幾何布朗運動下，並假設利率與房屋價值具有相關性，定價年金型RM商品。若未來屋主死亡時，累積的年金超過當期房屋價值造成銀行的損失取期望折現值後，與保險公司做為第三方擔保所收取的保險費取期望折現值後，令其兩者相等，在立體樹上求出公平的保險費率或可貸成數。此篇考慮三種情境，一是考慮借款人並無繼承人，須滿足「貸款人損失=貸款保險金+房屋殘值」；二是考慮借款人有繼承人，繼承人有還款或交付房屋的選擇權，須滿足「貸款人損失=貸款保險金」；三是只考慮貸款人損失與房屋殘值，而期間不繳交保險費，須滿足「貸款人損失=房屋殘值」。此篇發現情境一與三所繪出的圖形為線性，但情境二卻是非線性，原因是情境二中的貸款人損失可看作是一賣權，其標的物為房屋，履約價為貸款總額，而選擇權的報酬是非線性的，而在情境一與三下，房屋殘值可看作成買權，當貸款人損失減去房屋殘值時，可看作是一賣權減去買權，其方程式可仿照put-call partity改寫成累積貸款總額減房屋價值，故所繪出的圖形是線性的。本篇一開始假設期初不須繳保險費，但在情境二下若給定保險費率則發現可貸成數無解，因為若只增加保險費率之下，同時會增加貸款人損失，且計算出的貸款保險金會因為是以每期年金的比例做計算，而使方程式最終會是一個可貸成數×保險費率×初始房價的函數式，而得出的貸款保險金並不會大於或等於貸款人損失，故在期初增加房價的2%作為保費，即可解出可貸成數，但可貸成數算出的值較其他情境低，發現是因貸款人的繼承人在到期時有選擇權可決定還款或者給予房屋，此權力造成銀行的損失風險增加，才使可貸成數較低。

Burden on national economies are becoming heavier due to the costs of social welfare programs, like health care and annuities, for the increasing elder population. To address this problem, a new product “reverse mortgage” (RM), that allows an elder home owner (or the borrower) to pledge his real estate to the bank for periodical incomes without losing the using right. This thesis focuses on the pricing of perpetual annuity type of RM. My pricing model incorporates the Hull-White interest rate model, Lee-Carter morality model, geometric Brownian motion house price model, and the correlation between the interest rate and the house price. The bank receives the estate as the borrower died but cannot demand extra payments even if the cumulative loan amount exceeds the value of the estate. In return, the bank can charge the mortgage insurance premium. My model can calculate the fair values of the lender loss and the insurance premium by the risk neutral valuation method— calculating the expected discounted values of potential loss and premium under the risk-neural probability. The fair insurance fee (or the amounts of fair annuity payments) can be obtained by equating these two values. I consider three scenarios. First, if the borrower has no heirs, it must meet the equation "lender losses =insurance premium + residual house value"; second, if the borrower have heirs, there’s an option for heirs to choose repayment or pay house, must meet the "lender losses = Loan Insurance"; third, if borrower not to pay premiums during the life, it meet "lender loss = residual house value. The fair insurance premium rate (annuity amount) can be quantitatively evaluated by solving the root of the function f derived by moving aforementioned terms to the right hand side. The function f of the first and third scenarios is linear, but the f for the second scenario is non-linear. The reason is that the lender can be viewed as put options on the estate price. The strike price is total loan amounts. The nonlinear option payoff cause scenario 2 produces a convex f. On the other hand, the function f for scenario one and three contain the term lender losses minus residual house value, which is analog to a put option minus a call options. This value can be rearranged as total loans minus the house price, which is also linear. I assume that no upfront insurance premium is paid at the beginning. But the root of function f can’t be solved for fair annuity amount given the premium rate. If I only increase the premium rate, it will increase the losses of lender, and the insurance premium is calculated based on the ratio of annuity payments, so the equation is the function of annuity×premium rate×initial house price, and the insurance premiums will not be greater than or equal to lender losse, so I add 2% of house price as the upfront insurance premium to make f solvable. This is because the borrower's heirs own an option to choose whether to repay the loan or give up the house. This option increases the risk of banks, so the annuity amount in scenario 2 is lower than the other ones.