Introduction

In this chapter, we delve into the intricacies of calculating fixed-rate mortgage payments. Fixed-rate mortgages are one of the most common types of residential mortgage products, characterized by a stable interest rate over the life of the loan. Understanding how these payments are computed is essential for both borrowers and investors in mortgage-backed securities.

Fixed-rate mortgages are a popular choice for many homebuyers due to their predictability and long-term stability. The monthly payment for a fixed-rate mortgage includes both the principal repayment and the interest payment. The payment remains constant throughout the loan term, but the allocation between principal and interest changes over time as the outstanding balance decreases.

Calculating Fixed-Rate Mortgage Payments

The formula to calculate the fixed monthly payment for a fixed-rate mortgage is given by: $$ P M T=\frac{P \times r \times (1+r)^n}{(1+r)^n-1} $$ Where:

• $P$ is the principal amount of the loan.
• $r$ is the monthly interest rate (annual interest rate divided by 12 and expressed as a decimal).
• $n$ is the number of total payments (loan term in months).

Example: Consider a USD 200,000 fixed-rate mortgage with an annual interest rate of 4.5% and a 30-year term. First, convert the annual interest rate to a monthly rate, $r = \frac{0.045}{12} = 0.00375$. The number of total payments is: $n=30 \times 12=360$. Plugging these values into the formula:

$$PMT = \frac{200,000 \cdot 0.00375 \cdot (1 + 0.00375)^{360}}{(1 + 0.00375)^{360} - 1} \approx \text{USD 1,013.37}$$

So, the monthly payment for this mortgage is approximately USD 1,013.37, consisting of both principal and interest components.

Amortization Schedule and Component Allocation

An amortization schedule breaks down each monthly payment into its principal and interest components over the life of the loan. In the early years, a larger portion of the payment goes towards interest, while the principal portion gradually increases. As the loan matures, more of the payment is applied to the principal.