To calculate the balance of the loan after the first payment, we need to compute the monthly payment and apply it to the loan balance. The formula for the monthly payment of an amortized loan is:

$M = P \frac{r(1+r)^n}{(1+r)^n - 1}$

Where:

• $M$ = Monthly payment
• $P$ = Loan principal ($240,000)
• $r$ = Monthly interest rate (Annual rate / 12)
• $n$ = Total number of payments (Loan term in years × 12)

Step 1: Compute monthly payment

1. Loan principal ($P$) = $240,000
2. Annual interest rate = 6.80%, so monthly interest rate ($r$) = $6.80\% / 12 = 0.0056667$ (approx).
3. Loan term = 30 years = $30 \times 12 = 360$ months.

Substituting these into the formula for $M$: $M = 240,000 \frac{0.0056667(1+0.0056667)^{360}}{(1+0.0056667)^{360} - 1}$

Step 2: Calculate the monthly payment

Let me calculate this.# Loan parameters P = 240000 # Loan amount annual_rate = 6.80 / 100 # Annual interest rate monthly_rate = annual_rate / 12 # Monthly interest rate term_years = 30 # Loan term in years n = term_years * 12 # Total number of payments

Monthly payment formula

M = P * (monthly_rate * (1 + monthly_rate)**n) / ((1 + monthly_rate)**n - 1) M1564.6204515957831564.620451595783successThe monthly payment is approximately $M = 1564.62$.

Step 3: Compute the loan balance after the first payment

The balance after the first payment is calculated by determining how much of the payment goes toward interest and principal. The formula for the interest portion of the first payment is:

$\text{Interest} = \text{Previous Balance} \times \text{Monthly Rate}$

$\text{Principal} = M - \text{Interest}$

$\text{New Balance} = \text{Previous Balance} - \text{Principal}$

Substituting $\text{Previous Balance} = 240,000$, $M = 1564.62$, and $\text{Monthly Rate} = 0.0056667$, we calculate these values.# Calculate interest and principal for the first payment interest_first_payment = P * monthly_rate principal_first_payment = M - interest_first_payment

New balance after the first payment

new_balance = P - principal_first_payment interest_first_payment, principal_first_payment, new_balance(1360.0, 204.62045159578292, 239795.37954840422)(1360.0, 204.62045159578292, 239795.37954840422)successAfter the first payment:

• Interest paid: $1360.00
• Principal paid: $204.62
• New balance: $239,795.38

The loan balance after the first payment is approximately $239,795.38.

Would you like to see the detailed loan schedule or have any questions about the calculations?

Here are 5 follow-up questions to explore further:

1. How is the amortization schedule created for multiple payments?
2. What happens if Jennifer pays extra principal each month?
3. How does a shorter loan term (e.g., 15 years) affect the monthly payment?
4. How does a change in the interest rate impact the payment and total cost of the loan?
5. What are the tax implications of mortgage interest payments?

Tip: Understanding how the principal and interest portions evolve over time is key to planning prepayments effectively!