A company took a $1,000,000 loan from its local Bank at a rate of 5% compounded semi-annually but payable monthly over a term of 10 years. How many months will it take the company to pay off 50% of the loan? Thank you for help.
First, we have to convert 5% compounded semi-annually to compounded monthly. We can do that as follows: [1 + 0.05/2]^1/6 - 1 x 1,200 =4.94869855817...%.
Next, we can use this loan formula:
PMT=PV. R.{[1 + R]^N/ [1 + R]^N - 1}, Where R=Interest rate per period, N=number of periods, P=periodic payment, PV=Present value.
Now, will just substitute in the above formula as follows:
PMT =1,000,000 x 0.0494869855817/12 x [1 + 0.0494869855817/12]^(10*12) / [1 + 0.0494869855817/12]^(10*12) - 1]
PMT =4,123.91546514 x 2.56588515.......
PMT =$10,581.49 - This is the monthly payment of the loan.
Next, will use the following formula to solve for N.
-P*[(1-(1+R)^-N)/(R)]+FV*(1+(R))^-N-PV=0
If we plug in P=10,581.49, R=The above monthly interest rate, FV=500,000, and PV =1,000,000, then we solve for N:
N =67.33 months. In other words, on the 67th payment, half the loan, or about $500,000 will have been paid off.
