Question

has quoted her mortgage interest rate is 4.5%; this rate would be compounded semi-annually, while her payments would be made monthly. what is the effective monthly interest rate (emr) that she would pay?

Answer 1

The effective monthly interest rate (EMR) that she would pay is approximately
`\(0.0456\).`

Step-by-step explanation:

To find the effective monthly interest rate (EMR) for a mortgage with a nominal interest rate of 4.5%, compounded semi-annually, and monthly payments, we employ the formula:

`\[EMR = \left(1 + (r)/(n)\right)^n - 1\]`

where
`\(r\)` is the nominal interest rate, and
`\(n\)` is the number of compounding periods per year. In this scenario,
`\(r = 0.045\)` (4.5% expressed as a decimal) and
`\(n = 2\)` (compounded semi-annually).

Substituting these values into the formula, we get:

`\[EMR = \left(1 + (0.045)/(2)\right)^2 - 1\]`

Now, let's compute this:

`\[EMR = \left(1 + 0.0225\right)^2 - 1\]`

`\[EMR = 1.0225^2 - 1\]`

`\[EMR = 1.0456 - 1\]`

`\[EMR = 0.0456\]`

So, the effective monthly interest rate (EMR) is approximately
`\(0.0456\),`or
`\(4.56\%\).`

This calculation reveals that even though the nominal interest rate is 4.5%, the effective monthly rate, considering semi-annual compounding, is slightly higher. This outcome underscores the importance of understanding the impact of compounding frequency on the overall interest cost. Borrowers, by knowing the EMR, can make more informed decisions about their mortgage, as it reflects the true cost of borrowing on a monthly basis.

The complete question is:

"Sarah has quoted her mortgage interest rate as 4.5%, and this rate would be compounded semi-annually. However, she plans to make her mortgage payments on a monthly basis. What is the effective monthly interest rate (EMR) that Sarah would pay?"