Car loan interest rate is 7%
loan payment $600 month
36 months left on loan
If pay an addition $1000 with next regular $600 payment (due in a month), how much will it reduce the amount of time left to pay off the loan?
present value of remaining balance
= 600( 1 - 1.00583333^-36)/.00583333
= $19,431.88
you are paying an extra $1000
leaving a present value balance of $18,431.88
so ...
600( 1 - 1.00583333^-n)/.005833333 = 18431.88
1 - 1.005833333^-n = .179198811
1.00583333^-n = .820801188
log it
-n log1.00583333 =log .82080...
-n = -33.95
n = appr 34
So the time would be reduced by 2 months
check:
present value of 34 payments of 600
= 600( 1 + 1.00583333^-34)/.00583333
= 18455.74 , close to $18431 , remember we rounded off the 33.95 to 34
To calculate how much the additional payment of $1000 will reduce the amount of time left to pay off the loan, we need to understand the impact of the extra payment on the remaining principal balance.
First, let's calculate the interest paid on the regular monthly payment of $600. The interest is calculated as (loan balance) x (monthly interest rate). Since the loan balance is not provided, we need to calculate it based on the remaining months.
The formula to calculate the loan balance is as follows:
Loan balance = (monthly payment) x [(1 - (1 + interest rate)^(-remaining months)) / interest rate]
Using this formula, we can calculate the loan balance after 36 months as follows:
Loan balance = $600 x [(1 - (1 + 0.07) ^ (-36)) / 0.07] = $16,822.38
Now, let's consider the impact of the additional payment of $1000. By making this extra payment, the remaining balance on the loan will be reduced. Therefore, the new loan balance would be:
New loan balance = Loan balance - Additional payment = $16,822.38 - $1000 = $15,822.38
To calculate the new remaining time to pay off the loan, we need to recalculate the number of months based on the new loan balance using the same formula as before:
Remaining months = -[log(1 - ((monthly payment) x (interest rate) / New loan balance))] / log(1 + interest rate)
Remaining months = -[log(1 - ($600 x 0.07 / $15,822.38))] / log(1 + 0.07) ≈ 16.46 months
Rounding up to the nearest whole month, the additional payment of $1000 is estimated to reduce the time left to pay off the loan by approximately 17 months (from 36 to 16).