SOLUTION: Please help me solve this
A $120,000 home mortgage for 30 years at 7 1/2% has a monthly payment of $839.06. Part of the monthly payment goes toward the interest charge on the u
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Question 888531: Please help me solve this
A $120,000 home mortgage for 30 years at 7 1/2% has a monthly payment of $839.06. Part of the monthly payment goes toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that goes toward the interest is given by:
u equal M minus open parentheses M minus fraction numerator P r over denominator 12 end fraction close parentheses open parentheses 1 plus r over 12 close parentheses to the power of 12 t end exponent
and the amount that goes toward the reduction of principal is given by:
v equal open parentheses M minus fraction numerator P r over denominator 12 end fraction close parentheses open parentheses 1 plus r over 12 close parentheses to the power of 12 t end exponent
In these formulas, P is the size of the mortgage, r is the interest rate, M is the monthly payment, and t is the time (in years).
(a) Approximate the time when the monthly payment is is evenly divided between interest and principal reduction.
(b) Repeat the problem for a repayment period of 20 years (M = $966.71).
Thanks
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
i don't understand your formulas very well but i can tell you how to find the answer logically through the use of the annuity formula or through the use of a financial calculator.
your monthly payment is 839.06
if the principal and the interest are divided evenly, then the principal part of the payment must be half and the interest part of the payment must also be half.
half of 839.06 = 419.53
in order for the interest to be 419.53, then the remaining balance can be found as follows:
let x = remaining balance.
x = 419.53 / interest rate.
the interest rate is equal to .075 / 2 = .00625
x = 419.53 / .00625 = 67124. 8
you can calculate the number of months it takes to pay that off by using the annuity formula.
you have the same monthly interest rate of .00625.
you have a present value of 67124.8
you have a monthly payment of 839.06
you solve for the number of time periods to get 111.25
if it takes 111.25 more months to reduce that remaining balance to 0, then you must have already gone through 360 - 111.25 = 248.75 months to get that remaining balance.
that should be your answer.
you can use the formula or a calculator to get future value of a loan payment of 839.06 on a principal of 120,000 at a monthly interest rate of .00625 for 248.75 months and you will find that the future value of that loan is equal to 67124.8 or something very close.
when the remaining balance is 67124.8, the interest on that will be 67124.8 * .00625 = 419.53.
take that away from the payment of 839.06 and you get a principal of 419.53 as well.
the same procedure can be used for a loan of 20 years at the same interest rate with a monthly payment of 966.71
the manual calculations are arduous so i'll use the finanaial calculator.
20 years * 12 = 240 monthly payments.
annual interest rate of .075 / 12 = monthly interest rate of .00625.
monthly payment is 966.71
half of 966.71 = 483.355
x * .00625 = 483.355
solve for x to get:
x = 77336.8
your remaining balance has to be 77336.8 in order for the payment to be half interest and half principal.
77336.8 * .00625 = 483.355.
483.355 + 483.355 = 966.71
now you want to know how many months to pay off 77336.8 at .00625 interest per month with a payment of 966.71 per month.
that comes out to be 111.25 months
140 - 111.25 = 128.75 momnths.
the loan must have gone 128.75 months.in order for the payment to be half interest and half principal.
set number of time periods to 128.75 months
set interest rate to .00625 per month.
set payment to 966.71 per month
set present value to 120,000
solve for future value of a payment to get fv = 77337
77337 * .00625 = 483.356
double that to get 966.671
pretty close.
i'm pretty sure i could do the math if i had the exact formulas to work with but i couldn't make them out from what you showed.
sorry about that.
i do believe the answers i provided, however, are correct if i understood the problem correctly.
-------
i was subsequently able to figure out the formula for U and V.
the information is shown below from a followup memo that I sent you.
your original problem statement was:
A $120,000 home mortgage for 30 years at 7 1/2% has a monthly payment of $839.06. Part of the monthly payment goes toward the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that goes toward the interest is given by:
u equal M minus open parentheses M minus fraction numerator P r over denominator 12 end fraction close parentheses open parentheses 1 plus r over 12 close parentheses to the power of 12 t end exponent
and the amount that goes toward the reduction of principal is given by:
v equal open parentheses M minus fraction numerator P r over denominator 12 end fraction close parentheses open parentheses 1 plus r over 12 close parentheses to the power of 12 t end exponent
In these formulas, P is the size of the mortgage, r is the interest rate, M is the monthly payment, and t is the time (in years).
(a) Approximate the time when the monthly payment is is evenly divided between interest and principal reduction.
(b) Repeat the problem for a repayment period of 20 years (M = $966.71).
your formula for V looks like this:
V = (M-Pr/12)(1+r/12)^12t
V = amount that goes to reduction of principal.
M = monthly payment = 839.06
P = principal = 120,000
t = time in years.
r = interest rate per year.
If we want the mortage payment to be equally divided between amount that goes to reduction of principal and amount that goes towards interest, then you need to divide the monthly payment by 2.
839.06 / 2 = 419.53
your formula of:
V = (M-Pr/12)(1+r/12)^12t becomes:
419.53 = (839.06 - 120,000*.075/12)*(1.00625)^12t
simplify this to get:
419.53 = (89.06) * (1.00625)^12t
divide both sides of this equation by 89.06 and you get:
419.53/89.06 = 1.00625^12t
simplify to get:
4.710644509 = 1.00625^12t
take the log of both sides of this equation to get:
log(4.710644509) = log(1.00625^12t)
since log(1.00625^125) = 12t * log(1.00625), your equation becomes:
log(4.710644509) = 12t * log(1.00625)
divide both sides of this equatgion by log(1.00625) to get:
log(4.710644509) / log(1.00625) = 12t
solve for 12t to get:
12t = 248.7460656
since 12t is equal to the number of months, that's your answer.
your equation for amount that goes to the principal is equal to:
U = M - ((M-Pr/12)(1+r/12)^12t)
this is equivalent to U = M - V since V = (M-Pr/12)(1+r/12)^12t
so if the amount that goes to reduction of principal is equal to 419.53, then the amount that goes towards interest has to be equal to 839.06 - 419.53 which is equal to 419.53.
the bottom line is you only have to find V for M/2 which we did above.
U will be equal to M - V.
for your second problem, you use the same formula of:
V = (M-Pr/12)(1+r/12)^12t
in this case:
M = 966.71
V = M/2 = 483.355
P = 120,000
r = .075
your formula becomes:
483.355 = (966.71 - 120,000 * .075/12) * (1.00625)^12t
simplify to get:
483.355 = 216.71 * 1.00625^12t
divide both sides of this equation by 216.71 to get:
483.355 / 216.71 = 1.00625^12t
simplify to get:
2.230423146 = 1.00625^12t
take the log of both sides of this equation to get:
log(2.230423146) = log(1.00625^12t)
since log(1.00625^12t) = 12t * log(1.00625), your equation becomes:
log(2.230423146) = 12t * log(1.00625)
divide both sides of this equation by log(1.00625) to get:
log(2.230423146) / log(1.00625) = 12t
solve for 12t to get:
12t = 128.7512902
your principal reduction amount and interest will both be equal to 483.355 in 128.7512902 months from the start of the loan.
there is an online financial calculator that you can use to take the drudgery out of the calculations.
that calculator can be found at the following link:
http://www.arachnoid.com/lutusp/finance.html
the thing to remember is that, if PV is positive, then PMT needs to be negative.
same goes with PV and FV.
if PV is negative, then FV must be positive.
this assume you are entering PV or PMT or FV and asking the calculator to get you either the interest rate or the number of time periods.
the interest rate is entered as a decimal, i.e. 7.5 is entered at .075.
you need to tneter the interest rate per time period, so if the time period is in months, then the annual interest rate is divided by 12 and the number of time periods is multiplied by 12 if it is in years.
enjoy.
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