Vanna has just financed the purchase of a home for $200 000. She agreed to repay the loan by making equal monthly blended payments of $3000 each at 4%/a, compounded monthly
a. How long will it take to repay the loan?
b. How much will be the final payment?
c. Determine how much interest she will pay for her loan.
i = .04/12 = .00333... (I store this in my calculator's memory for maximum accuracy)
n is number of months, so
200,000 = 3000 (1 - 1.003333...^-n)/.003333...
.2222.... = 1 - 1.00333...^-n
1.003333.^-n = .77777...
take log of both sides and use log rules
-n log1.003333... = log .7777..
-n = log .777.../log 1.003333.. = -75.519..
So you will need 75 monthly payments of $3000 and a partial payment at month 76.
b) balance at the end of 75 months
= 200000(1.0033333..)^75 - 3000(1.0033333...^75 - 1)/.0033333...
= 1555.81
interest on that amount for 1 months = .003333..(1555.81) = 5.186
final partial payment = 1555.81 + 5.186 = $1560.99
c) this is actually an invalid question, since you would be adding up sums of money that are not at the same time spot.
nevertheless, they probably want:
interest paid = total amount paid - 200,000
= 75(3000) + 1560.99 - 200000 = $26,560.99
check my arithmetic
To answer these questions, we can use the formula for the present value of an annuity:
PV = P * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value (loan amount)
P = Monthly payment
r = Monthly interest rate
n = Number of payments (duration)
To find the answers, we will use this formula and solve for different variables.
a. How long will it take to repay the loan?
To find the duration, we need to solve for 'n'. We already have the values for PV ($200,000), P ($3,000), and r (4% per year compounded monthly).
PV = P * (1 - (1 + r)^(-n)) / r
Plugging in the values:
$200,000 = $3,000 * (1 - (1 + 0.04/12)^(-n)) / (0.04/12)
Simplifying the equation and solving for 'n':
(1 - (1 + 0.04/12)^(-n)) = $200,000 * (0.04/12) / $3,000
Solving using logarithms, we find that 'n' is approximately 55.12 months.
Therefore, it will take Vanna approximately 56 months (rounding up) to repay the loan.
b. How much will be the final payment?
The final payment can be calculated by subtracting the total amount already paid from the loan amount.
Total amount paid = P * n
Substituting the known values:
Total amount paid = $3,000 * 56
The total amount paid is $168,000.
Final payment = Loan amount - Total amount paid
Final payment = $200,000 - $168,000
The final payment will be $32,000.
c. Determine how much interest she will pay for her loan.
The total interest paid can be obtained by subtracting the loan amount from the total amount paid.
Total interest paid = Total amount paid - Loan amount
Total interest paid = $168,000 - $200,000
The total interest paid will be -$32,000 (negative indicates that Vanna paid $32,000 less than the loan amount). Alternatively, you could think of the negative value as indicating that Vanna saved $32,000 in interest compared to not making any payments and repaying the loan over the same period.