Mortgage Payment:

Question

The equation below helps us determine the monthly payment for a mortgage.

𝑃
𝑎
𝑦
𝑚
𝑒
𝑛
𝑡
=
𝑃

⋅
(
𝑟
𝑛
)
1
−
(
1
+
𝑟
𝑛
)
−
𝑛
𝑡
Payment=
1−(1+
n
r
​
)
−nt

P ⋅(
n
r
​
)
​


P - principal (the borrowed amount)

n - the number of times the loan is compounded in a year

t - the number of years

r - the interest rate but written as a decimal

Example: A mortgage of $500,000 at 4.1% interest for 30 years would have a monthly payment of $2415.99.

John is going to make a house purchase and borrow
$
325
,
000
$325,000 and has two options.

John can have a mortgage for
30
30 years an interest rate of
4.5
%
4.5% OR a mortgage for
15
15 years at an interest rate of
4.25
%
4.25%

What is
𝑟
r for the
30
30 loan?


give me a step by step

Answer 1

To determine \( r \) for the 30-year mortgage loan with an interest rate of 4.5%, we need to convert the percentage into a decimal. Here's how to do it step by step:

Step 1: Understand the Interest Rate

The interest rate given is 4.5%. This is an annual interest rate.

Step 2: Convert the Percentage to Decimal

To convert a percentage to a decimal, divide by 100:

\[ r = \frac{4.5}{100} = 0.045 \]

Step 3: Understand the Compounding Frequency

The problem states the loan is a mortgage. Typically, mortgage loans in the United States are compounded monthly. This means \( n = 12 \) (the number of compounding periods per year).

Step 4: Find the Monthly Interest Rate

To find the monthly interest rate, you need to divide the annual interest rate \( r \) by the number of compounding periods in a year \( n \):

\[ r_n = \frac{r}{n} = \frac{0.045}{12} \]

Step 5: Calculate \( r_n \)

Now calculate \( r_n \):

\[ r_n = \frac{0.045}{12} \approx 0.00375 \]

Summary

Thus, the value of \( r \) for the 30-year loan is:

\[ r = 0.045 \quad (\text{annual interest rate as a decimal}) \] And the monthly interest rate \( r_n \) is:

\[ r_n \approx 0.00375 \]

This is the value you would use in the mortgage payment formula.