## Application Fixed Rate Mortgage Payments

Mortgages are annuities, so the annuity formula is quite useful.

Most mortgages are fixed rate mortgage loans, and they are basically annuities. They promise a specified stream of equal cash payments each month to a lender. A 30-year mortgage with monthly payments is really a 360-payments annuity. (The "annu-ity" formula should really be called a "month-ity" formula in this case.) What would be your monthly payment if you took out a 30-year mortgage loan for $500,000 at an interest rate of 7.5% per annum?

Before you can proceed further, you need to know one more bit of institutional knowledge here: Mortgage providersâ€”like banksâ€”quote interest by just dividing the mortgage quote by 12, so the true monthly interest rate is 7.5%/12 = 0.625%. (They do not compound; if they did, the monthly interest rate would be (1 + 7.5%)1/12 - 1 = 0.605%.)

A 30-year mortgage is an annuity with 360 equal payments with a discount rate of 0.625% per month. Its PV of $500,000 is the amount that you are borrowing. You want to determine the fixed monthly cash flow that gives the annuity this value:

Lenders quote interest rates using the same convention that banks use.

The mortgage payment can be determined by solving the Annuity formula.

Solving this for the cash flow tells you that the monthly payment on your $500,000 mortgage will be $3,496.07 for 360 months, beginning next month.

Uncle Sam allows mortgage borrowers to deduct the interest, but not the principal, from their tax bills. The IRS imputes interest on the above mortgage as follows: In the first month, Uncle Sam proclaims 0.625%-$500,000 = $3,125 to be the tax-deductible mortgage interest payment. Therefore, the principal repayment is $3,496.07 - $3,125 = $371.07 and remaining principal is $499,628.93. The following month, Uncle Sam proclaims 0.625%-$499,628.93 = $3,122.68 to be the tax-deductible interest payment, $3,496.07 - $3,122.68 = $373.39 as the principal repayment, and $499,255.54 as the remaining principal. And so on.

SIDE NOTE

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