# Fixed-Rate Mortgage

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## Definition: Fixed-Rate Mortgage

A fixed-rate mortgage is a fully amortizing loan where the interest rate on the mortgage is fixed throughout the term of the loan. This fixed rate as well as the term of the loan is known at the beginning of the loan. This provides the benefit of consistent and fixed payments of the principle and interest payments, allowing the borrower to plan periodic budgets by treating the fixed periodic payment as fixed costs. The borrower therefore will not have to worry about varying loan payment amounts due to fluctuating interest rates.

For a fixed-rate mortgage, a part of the periodic loan payment is to pay off the principal while the remainder constitutes the interest payment. The interest payment is calculated as the interest rate times the principal amount due at the beginning of the period. As a part of the principal is paid back each period, the interest component of the loan payment that is calculated on the principal, due at the beginning of the period is smaller in each period. Thus, in the initial payments, most of the payment contributes to interest payment while in later periods a major part of the loan payment goes towards the principal.

The interest rate on fixed-rate mortgages are higher than on floating/flexible rate mortgages as a result of the lower risk-level involved, making the mortgage expensive when interest rates fall.

If P is the fixed-rate mortgage loan that will be fully amortized by a fixed monthly payment M, the principal amount due after 1 period is given by: P1 = P + i*P – M or P * (1 + i) - M; where i is the fixed rate of interest.

The principal due after the second period is P2 = P1 * (1 + i) – M.

Substituting for P1, P2 can be written as P2 = P*(1 + i)2 – M(1 + i) – M.

Proceeding on similar lines, a generalized expression for the principal due at the end of the Nth period can be expressed as PN = P*(1 + i) – M*{(1 + i)N-1+(1 + i)N-2....+1}.

Since the principal amount is completely amortized by the Nth payment, PN will be zero.

This yields the monthly payment amount to be:

M = P * (i* (1+i)^N)/((1+i)^N- 1)

Example:

Suppose the interest rate on a 20 year fixed-rate mortgage of \$100,000 is 10%. The Monthly payment can be calculated as M = \$965 that will fully amortize the loan in 240 payments.

Hence, this concludes the definition of Fixed-Rate Mortgage along with its overview.

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