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Today's mortgage industry is constantly changing, with adjustable rate mortgages (ARM), loans originated to the so-called "subprime" market, and volatile interest rates. Amid the changes and controversy, lenders continue to originate loans because the interest paid over the loan lifetime is profitable. Measuring the profitability of those loans, along with return on investment to the lender is assessed using Actuarial Present Value (APV), which incorporates the uncertainty that exists in the mortgage industry today, with many loans defaulting and prepaying. The hazard function, or instantaneous failure rate, is used as a measure of probability of failure to make a payment. Using a logit model, the default and prepayment risks are estimated as a function of interest rate. The "optimal" interest rate can be found where the profitability is maximized to the lender.

房屋抵押貸款的評價，因為需考慮到貸款人的提前清償及違約風險造成現金流量之不確定性，決定房屋抵押貸款的價格比決定一般違約證券的價格更具困難度。因此，如何合理評估抵押貸款證券的價值實為一值得深入探討之課題，本文即針對此議題進行研究。傳統文獻在進行房屋抵押貸款的評價方法，主要可區分為兩種：結構模型(structural-form approach)及縮減式模型(reduced-form approach)。目前的文獻上，其評價的封閉解只存在於結構式模型，但在此模型下的評價，存在著違約與提前清償條件的設定問題，這將對評價的準確性造成很大的影響，在實務的應用上有一定的限制。再者，結構式模型在處理多變數且變數間具相關性的情況，存在一定的複雜性與困難度，而縮減式模型在此情況的處理上是較容易的。本研究將從縮減式模型的角度，引入 Jarrow (2001)的概念，在包含多重變數並考慮變數間相關係數之縮減模型下，進行房屋抵押貸款封閉解的推導。透過此方法可協助資產管理者從事投資組合配置最適化與避險策略的分析，亦期望能提供實務界一個更具可行性與效率性之房屋抵押貸款評價模型。 / Valuing mortgage-related securities is more complicated than valuing regular defaultable claims due to the borrower’s prepayment behavior as well as the possibility of default. In general, the methods that are applied to investigate mortgage value and termination risk can be divided into two categories: a structural-form approach and a reduced-form approach. Some researchers use a structural-form model to obtain the closed-form formulae for the mortgage value. With this method, however, it is difficult to identify the critical region of early exercise and deal with the situation including multivariable and their correlations correlation among variables. As an alternative, the reduced-form model developed in this study is able to value the mortgage without setting boundary conditions, and can thereby accurately handle the multi-dimensional space of correlated state variables. This study extends Jarrow’s (2001) model to examine mortgage valuations. The purpose of this article is to derive a closed-form solution of the mortgage valuation equation under a general reduced-form model that embeds relevant economic variables. This new approach enables portfolio managers to undertake sophisticated portfolio optimization and hedging analyses, and makes it possible to more accurately and efficiently value the complicated mortgage.