Question

For a loan of $100,000, at 4 percent annual interest for 30 years, find the balance at the end of 4 years assuming monthly payments.

Answer 1

To find the remaining loan balance after 4 years, you must first calculate the monthly payment for a $100,000 loan at 4 percent annual interest over 30 years. Then use this monthly payment and the number of payments made to compute the balance using the appropriate annuity formula.

Step-by-step explanation:

Finding the Remaining Balance on a Loan

To find the balance at the end of 4 years on a $100,000 loan at 4 percent annual interest with a 30-year term and monthly payments, we need to first calculate the monthly payment amount and then use that figure to determine the outstanding balance.

First, we calculate the monthly payment using the formula for an annuity:

PMT = P * (r(1+r)^n) / ((1+r)^n - 1)

Where:

PMT is the monthly payment

P is the principal amount of the loan ($100,000)

r is the monthly interest rate (0.04/12)

n is the total number of payments (30 * 12)

Once we have the monthly payment, we will then calculate the outstanding balance after 4 years (or 48 payments) using the remaining balance formula for an annuity:

B = P * ((1+r)^n - (1+r)^p) / r

Where:

B is the balance

P is the original principal amount

r is the monthly interest rate

n is the total number of payments

p is the number of payments made (48)

Answer 2

The balance at the end of 4 years for a loan of $100,000 at 4% annual interest for 30 years, assuming monthly payments, is approximately $93,773.03.

Step-by-step explanation:

To find the balance at the end of 4 years for a loan of $100,000 at 4% annual interest for 30 years and assuming monthly payments, we can use the formula for calculating the balance on a loan after a certain number of periodic payments:

Balance = Loan Amount * (1 + Monthly Interest Rate)^Number of Payments − Monthly Payment * [((1 + Monthly Interest Rate)^Number of Payments) − 1] / Monthly Interest Rate

Plugging in the values, we have:

Loan Amount = $100,000, Monthly Interest Rate = (4% / 12) = 0.00333, Number of Payments = 4 * 12 = 48, Monthly Payment = ?

Solving for the Monthly Payment using the formula above, we get:

Monthly Payment = $481.33

Now, we can substitute the values back into the formula to find the balance at the end of 4 years:

Balance = $100,000 * (1 + 0.00333)^48 − $481.33 * [((1 + 0.00333)^48) − 1] / 0.00333

Simplifying the calculation, we find that the balance at the end of 4 years is approximately $93,773.03.