Thursday, June 30, 2005
compount interest and refinancing problem.
i came across a math website explaining the component interest and adv on refinance...
Example: If you invest $2500 at 9% annual interest, compounded quarterly (four times a year), how much money will you have after 10 years?
Solution: We use the compound interest formula:
C = P(1 + r/n)^nt = 2500(1 + 0.09/4)40 = $6087.97.
Example: You borrow $9634.67 at 9% interest, compounded monthly and plan to make the minimum payments of $200 at the end of each month for five years. After two years, what is the payoff amount on the loan?
Solution: After 2 years, there are 5 - 2 = 3 years remaining on the loan, so that payoff amount is:
L = 200(1 - (1 + 0.09/12)-12*3)/(0.09/12) = 6289.36105
or $6289.36
Example: You borrow $200,000 at an annual interest rate of 6.5%, compounded monthly, to purchase a house. You plan on repaying the loan over the course of 30 years and make the minimum payments each month for 10 years, at which time you have the opportunity to refinance the loan at an interest rate of 5.75%. What are the minimum payments on the new loan (assuming you still wish to pay off the loan 30 years after you borrowed the $200,000) and how much money will you save by refinancing?
Solution: We first use the loan formula to find the minimum payments for the original loan:
200000 = P(1 - (1 + 0.065/12)-12*30)/(0.065/12)
which we solve to get P = 1264.136047, or $1264.14.
After 10 years of making monthly payments of $1264.14, there will be 20 years remaining on the original loan. To find the payoff amount we use t = 20 in the loan formula with P = 1264.14, r = 0.065 and n = 12:
L = 20000(1 - (1 + 0.065/12)-12*20)/(0.065/12) = 169552.7829
so the payoff amount is $169,552.78.
So we want to take out a new loan for $169,552.78 at an annual interest rate of 5.75%, compounded monthly, and we wish to repay this new loan over the course of 20 years. We use the loan equation for a third time to find the minimum payments required under the new loan:
169552.78 = P(1 - (1 + 0.0575/12)-12*20)/(0.0575/12)
to get P = 1190.402106, so the new monthly payments will be $1190.40.
This is a remarkably better situation, since for the final 20 years of the loan (or 240 payments) we will only need to pay $1190.40 instead of $1264.14, a savings of $1264.14 - $1190.40 = $73.74 a month. Over the course of 20 years of monthly payments this amounts to a savings of $73.74*12*20 = $17,697.60.
There are often fees associated with refinancing, which can range from a few hundred dollars to several thousand dollars. In this case an investment of $1000, say, would be worthwhile in order to save over 17 times that amount over time.
It should also be noted that when refinancing you will likely have the option of taking out a new 30-year loan, but it makes sense to plan to repay the new loan over the span of time remaining on the original loan. First of all, it will save you more money; secondly, who wants to be making house payments for 40 years?
Example: If you invest $2500 at 9% annual interest, compounded quarterly (four times a year), how much money will you have after 10 years?
Solution: We use the compound interest formula:
C = P(1 + r/n)^nt = 2500(1 + 0.09/4)40 = $6087.97.
Example: You borrow $9634.67 at 9% interest, compounded monthly and plan to make the minimum payments of $200 at the end of each month for five years. After two years, what is the payoff amount on the loan?
Solution: After 2 years, there are 5 - 2 = 3 years remaining on the loan, so that payoff amount is:
L = 200(1 - (1 + 0.09/12)-12*3)/(0.09/12) = 6289.36105
or $6289.36
Example: You borrow $200,000 at an annual interest rate of 6.5%, compounded monthly, to purchase a house. You plan on repaying the loan over the course of 30 years and make the minimum payments each month for 10 years, at which time you have the opportunity to refinance the loan at an interest rate of 5.75%. What are the minimum payments on the new loan (assuming you still wish to pay off the loan 30 years after you borrowed the $200,000) and how much money will you save by refinancing?
Solution: We first use the loan formula to find the minimum payments for the original loan:
200000 = P(1 - (1 + 0.065/12)-12*30)/(0.065/12)
which we solve to get P = 1264.136047, or $1264.14.
After 10 years of making monthly payments of $1264.14, there will be 20 years remaining on the original loan. To find the payoff amount we use t = 20 in the loan formula with P = 1264.14, r = 0.065 and n = 12:
L = 20000(1 - (1 + 0.065/12)-12*20)/(0.065/12) = 169552.7829
so the payoff amount is $169,552.78.
So we want to take out a new loan for $169,552.78 at an annual interest rate of 5.75%, compounded monthly, and we wish to repay this new loan over the course of 20 years. We use the loan equation for a third time to find the minimum payments required under the new loan:
169552.78 = P(1 - (1 + 0.0575/12)-12*20)/(0.0575/12)
to get P = 1190.402106, so the new monthly payments will be $1190.40.
This is a remarkably better situation, since for the final 20 years of the loan (or 240 payments) we will only need to pay $1190.40 instead of $1264.14, a savings of $1264.14 - $1190.40 = $73.74 a month. Over the course of 20 years of monthly payments this amounts to a savings of $73.74*12*20 = $17,697.60.
There are often fees associated with refinancing, which can range from a few hundred dollars to several thousand dollars. In this case an investment of $1000, say, would be worthwhile in order to save over 17 times that amount over time.
It should also be noted that when refinancing you will likely have the option of taking out a new 30-year loan, but it makes sense to plan to repay the new loan over the span of time remaining on the original loan. First of all, it will save you more money; secondly, who wants to be making house payments for 40 years?
# posted by NDR @ 2:01 PM 
