A second mortgage was taken 10 years ago for $250,000 over 30 years at 6% compounded monthly. In addition to the regular monthly payments, $1,000 additional payments were made each and every year, on the anniversary date of the mortgage, and applied to the principal. The mortgage was sold recently for $243,023.18. What rate did the purchaser pay for this mortgage? Any help would be great. Thank you.
I will attempt to answer this question using this online financial calculator: https://arachnoid.com/finance/
1- The first thing to do is to calculate the monthly payments on the original mortgage of $250,000 @ 6% over 30 years or 360 months =$1,498.88
2 - Will convert the $1,000 per year applied to the principal to 12 equivalent monthly payments, using this formula: PMT = FV*(((1 + R)^N - 1)^-1* R), which comes to =$81.07, which will add to the regular payment in (1) above and we get: $1,498.88 + $81.07 =$1,579.95. This is the payment that will be used to reduce the mortgage to its balance prior to its sale.
3 - Applying the above monthly payment of $1,579.95 for 10 years, or 120 months, brings the mortgage down to its current balance, prior to its sale, to: $195,928.01
4 - Now that we have 4 known variables on this mortgage, Its FV, PV, PMT, and N, we can solve for the 5th, namely the interest rate at which it was purchased - by plugging all of them into this consolidated TVM formula:
-PMT*[(1-(1+R)^-N)/(R)]+FV*(1+(R))^-N+PV=0
5 - The above formula is programmed into my computer, which has the Newton-Raphson iteration and interpolation formula programmed into it and can easily and rapidly arrive at the answer, which is =7.375% =7 3/8%.
AND THAT IS THE END!.
