Abstract
In this paper, an Adjustable Rate Mortgage (ARM) and a Fixed Rate Mortgage (FRM) are formalized and studied in a simple continuous-time setting under the assumption of a simple one-factor Affine Term Structure (ATS). Through an application of existing results from ATS theory, it is shown that when the short rate reaches a certain pre-determined boundary, the constant payment stream on a new FRM equals the payments on an existing ARM. Hereby, this paper provides a theoretical build-in cap on the formalized ARM. The finite boundary for the short-rate suggests that certain caps on ARMs should (in theory) be offered free of charge.
Highlights
In this paper, the relation between the payments on a formalized Adjustable Rate Mortgage (ARM) and a Fixed Rate Mortgage (FRM) is studied in a simple continuous-time setting under the assumption of a simple one-factor Affine Term Structure (ATS)
Through an application of existing results from ATS theory, it is shown that when the short rate reaches a certain pre-determined boundary, the constant payment stream on a new FRM equals the payments on an existing ARM
The theorems and definitions summarized here are presented in a version which only applies to the simple affine short rate model without jumps presented in Definition 1
Summary
The relation between the payments on a formalized Adjustable Rate Mortgage (ARM) and a Fixed Rate Mortgage (FRM) is studied in a simple continuous-time setting under the assumption of a simple one-factor Affine Term Structure (ATS). The conclusion relies on an assumption of a one-factor ATS, and is a direct consequence of existing results provided by Keller-Ressel and Steiner [1] regarding the yield curve shapes of ATSs. The two mortgage products are formalized following the work of Nordfang and Steffensen [2]. It is shown how the boundary at which the interest rate can be fixed is at a higher level than the initial FRM interest This points out shortcomings of the usual budget assessment of ARM borrowers in the Danish mortgage market today.
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