personal finance

Question

Rounding in the calculation of monthly interest rates is discouraged. Such rounding can lead to answers different from those presented here. For long-term loans, the differences may be pronounced.

Several years ago Bill got a home mortgage of $123,000 with a term of 30 years at an APR of 9%. Use the Monthly Payments for Long-Term Loans Rule of Thumb to estimate his monthly payment.

$  922.50

Compare this estimate with what the monthly payment formula gives. (Round your answer to the nearest integer.)

The actual monthly payment is about $  ----  greater than the estimated payment.

Answer 1

Hi Christine!

You're already on the right track! Your estimated monthly payment of $922.50 is correct using the rule of thumb.

Now we just need to calculate the actual monthly mortgage payment and compare the two amounts.

Let's work through it step by step!

Step 1: Identify the information

We're given:

1. Mortgage principal (P) = $123,000
2. Annual interest rate (APR) = 9%
3. Loan term = 30 years
4. Estimated monthly payment = $922.50

Since mortgage payments are made monthly, we need to convert the annual interest rate and loan term into monthly values.

Step 2: Calculate the monthly interest rate

Divide the annual interest rate by 12:

[

r=\frac{0.09}{12}

]

[

\boxed{r=0.0075}

]

This represents a monthly interest rate of 0.75%.

Step 3: Calculate the total number of payments

The mortgage lasts 30 years, with 12 payments each year.

[

n=30\times12

]

[

\boxed{n=360}

]

Therefore, Bill will make 360 monthly payments.

Step 4: Apply the monthly payment formula

For a fully amortizing fixed-rate mortgage, the monthly payment formula is:

[

M=P\left[\frac{r(1+r)^n}{(1+r)^n-1}\right]

]

Where:

1. (M) = Monthly payment
2. (P) = Original loan amount
3. (r) = Monthly interest rate
4. (n) = Total number of payments

Substituting our values:

[

M=123,000\left[\frac{0.0075(1.0075)^{360}}{(1.0075)^{360}-1}\right]

]

Evaluating:

[

\boxed{M\approx$989.69}

]

This is the actual monthly principal-and-interest payment.

Step 5: Compare the actual payment with the estimate

The question asks how much greater the actual payment is than the estimated payment.

Subtract:

[

989.69-922.50=67.19

]

Round to the nearest whole dollar:

[

\boxed{$67}

]

Final Answer

The actual monthly payment is approximately $67 greater than the estimated monthly payment.

CalculationAmount
Rule-of-thumb estimate	$922.50
Actual monthly payment	$989.69
Difference	$67.19
Rounded answer	$67

Why Are the Two Payments Different?

The rule of thumb provides a quick estimate based on the loan's monthly interest.

However, the actual mortgage payment must cover both interest and repayment of the principal over the 30-year loan term.

The full amortization formula accounts for both components, which is why it produces a higher payment.

Helpful tip: Avoid rounding the monthly interest rate before performing the calculation. Even small rounding differences can affect mortgage calculations, particularly for long-term loans.

Hope this helps!

Michael