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18b_mortgage - Econ 251 Fall 2001 Mortgage Prepayments and...

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Econ 251 Fall 2001 Mortgage Prepayments and Valuation 1 Mortgage Arithmetic A mortgage is a promise backed by a house as collateral. There are records of mort- gages as far back as 1000 BC in Babylon. 1.1 Amortizing Mortgages Mortgages since the depression of the 1930s amortize. Unlike the old mortgages (or bonds), which pay a small coupon every period and a large fi xed sum at maturity, an amortizing ( fi xed rate) mortgage makes the same level payment every period. After the depression of the 1930s, in which many farmers defaulted on their loans just before the big fi nal payment came due, the amortizing loan became common. Mortgages had originally been structured like bonds, with a coupon payment covering the interest and then the big principal payment at maturity, because that made it easy to fi gure out how much a homeowner who wanted get out of his loan should pay. That is an important consideration, since nobody would ever borrow on the condition that he never move, and if the homeowner moves and sells his house, he no longer has the collateral to back the remainder of the loan. Under the old ”bullet” mortgage, it was easy to fi gure out what to charge the homeowner. No matter when he sold, he would have to make the next period payment and then give back the loan original loan amount (say $100). When the payment is not directly broken up into interest payments and a fi nal principal payment, it is not as easy to fi gure out what the homeowner should pay. But it is not so di cult either. 1.1.1 Computing the Level Payments If the mortgage rate is m , and the maturity of the mortgage is T periods, then the level payment x needed on a dollar loan of B 0 is fi gured out to solve the equation: B 0 = T X t =1 x 1 (1 + m ) t From our formula for annuities, this means that x = mB [1 1 (1+ m ) T ] Thus when T = , the mortgage pays x = m per period on a loan of B = $1 . If T = 30 and m = 7% and B = 1 , then (by our doubling rule) (1.07) 30 = 8 , and so x 8 7 m. 1
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A mortgage convention Typically, mortgage payments come monthly, and the mortgage rate is quoted at the monthly rate times 12. Thus an 8% mortgage means that T = 360 and m = 8 12 % . For simplicity (i.e., to fi t everything into an excel spread sheet) we will suppose that T = 30 years and m is an annual rate, and that all payments come at the end of the year. 1.1.2 Remaining Balance Just after the level payment x at any time n , the remaining balance is B n = T n X t =1 x 1 (1 + m ) t For example, suppose B = 100 , and m = 7% . Then the fi rst few years of level payments and balances work out to 8 . 06 8 . 06 8 . 06 8 . 06 8 . 06 8 . 06 100 98 . 94 97 . 81 96 . 60 95 . 30 93 . 91 92 . 43 1.2 Principal and Interest In keeping with the original form of the mortgage, which divided the payments into interest and principal, we can do the same with an amortizing mortgage. Notice that by construction, for every n , B n = 1 1 + m [ x + B n +1 ] Hence B n [1 + m ] = x + B n +1 x = payment = interest + principal paydown = mB n + ( B n B n +1 ) The division of the payment x between the interest payment mB n and the principal paydown amortizing amount ( x mB n ) = ( B n B n +1 ) starts o ff mostly interest and
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