反向房貸之隨機利率與模型風險

本文探討反向房貸商品，在估計借款人於契約終止時所能收到的償還金額，將會受到隨機利率的影響。由於反向房貸商品設計上，擁有無追索權的設計，若是在契約終止時，房子價值高於貸款價值，剩餘價值將會還給貸款人或其法定繼承人，反之，若房子價值低於貸款價值，則僅需支付貸款金額給借款人，因此借款人於契約終止時，所能收到的款項將會受到長壽風險、房價風險以及利率風險三項因素干擾，傳統在估計借款人所能收到之償還金額時，往往以長期利率水平做折現的動作，本文以傅立葉轉換定理，推導在Vasicek與CIR隨機利率模型底下，借款人所能收到之償還金額，迥異於直接用固定利率折現的結果，並且比較Vasicek與CIR隨機利率模型的結果，發現若是房價與利率存在相關係數，所能收到之償還金額也不相同，隨機利率模型的選取存在著模型風險。

關鍵字

反向房貸；隨機利率；利率風險；傅立葉轉換

並列摘要

This paper talks about the reverse mortgage program in the United States called Home Equity Conversion Mortgage (HECM). We come up with a method for calculating the repayment the lenders can get, and further find out that the repayment will be influenced by the stochastic interest rate model. Since the HECM is a non-recourse debt. When the loan terminates, if the property value is sufficient to pay the outstanding loan balance, the remaining value usually flows to the borrower or his/her beneficiaries. If the proceed is not enough to cover the loan balance, the non-recourse provision prevents the lender from pursuing other assets belonging to the borrower. So the repayment will be influenced by three major risks. The first one is the longevity risk, and the second one is the house price risk, and the final one is the interest rate risk. In conventional way of computing the repayment, the earlier method will discount the future cash flows by using long-term mean of interest rate. In this paper, we derive a new method for computing the repayment by using Fourier Transform under Vasicek and CIR model. We discover that the repayment will be different from the results by conventional way, and we further discover that if the house price and the interest rate have correlation, we also have interest rate model risk when computing the repayment.

並列關鍵字

Reverse Mortgage ； Stochastic Interest Rate ； Interest Rate Risk ； Fourier Transform

參考文獻

Chan, K., Karolyi, A. G., Longstaff, F. A., & Sanders, A. B. (1992). An Empirical Comparison of Alternative Models of the Short-Term Interest Rate. The Journal of Finance , pp. 1209-1227.

Chen, H., Cox, S. H., & Wang, S. S. (2010). Is the Home Equity Conversion Mortgage in the United States sustainable? Evidence from pricing mortgage insurance premiums and non-recourse provisions using the conditional Esscher transform. Insurance: Mathematics and Economics , pp. 371-384.

Chinloy, P., & Megbolugbe, I. F. (1994). Reverse Mortgages: Contracting and Crossover Risk. Real Estate Economics , pp. 367-386.

Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985). A THEORY OF THE TERM STRUCTURE OF INTEREST RATES. Econometrica , pp. 385-407.

Heston, S. L. (1993). A closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies , pp. 327-343.

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反向房貸之隨機利率與模型風險

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