APPENDIX TO CHAPTER 15Mortgages and Mortgage‐Backed Securities

A15.1 MONTH‐END BALANCES

This section shows that, under the principal amortization schedule described in the text, the balance outstanding at any time equals the present value of the remaining payments at the original mortgage rate. Let upper N be the term of the mortgage, in months; let r be the mortgage rate; let upper X be the monthly payment, and let upper B left-parenthesis i right-parenthesis be the balance outstanding at the end of month i, i equals 0 comma 1 comma ellipsis upper N. By definition, as discussed in the text,

(A15.1)upper B left-parenthesis 0 right-parenthesis equals upper X StartFraction 12 Over r EndFraction left-bracket 1 minus left-parenthesis 1 plus StartFraction r Over 12 EndFraction right-parenthesis Superscript negative upper N Baseline right-bracket

If the balance outstanding at the end of month i does equal the present value of the remaining payments at the rate , then,

According to the logic of the amortization table, ...

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