Question 1085648: Annuity Question.
fauzi buys a condominium for 80000 and he pays 10% down payment . the balance is to be paid in equal monthly instalments for thirty years. the first instalment is due one month after the date of purchase and the interest rate is 4.2%.compounded monthly.
A)if fauzi fails to make the first three monthly payments how much should he pay on the fourth payment to settle all the outstanding arrers?
B) immediately after paying for twenty years, Fauzi wants to settle the loan in full.how much is the amount that needs to be paid?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! he buys the condo for 80,000
he pays 10% down
the balance is to be paid in equal monthly installments for 30 years.
the first installment is due one month after the date of purchase and the interest rate is 4.2% compounded monthly.
if you use a financial calculator, like the Texas Instruments BA II Plus, then you need to do the following.
calculate the number of months.
this is equal to 30 years * 12 months per year for a total of 360 months.
calculate the interest rate percent per month.
this is equal to 4.2% / 12 = .35% per month.
calculate the remaining amount owed on the loan.
this is equal to 90% of 80,000 = 72,000
set future value equal to 0.
have the calculator tell you what the monthly payments need to be.
the calculator will tell you that the monthly payments are $352.0923651.
if you miss the first 3 payments, then you owe the bank the future value of those 3 payments plus interest to carry that future value to the end of the fourth month.
the future value of those first 3 payments are calculated as follows:
enter 3 for the number of periods
keep the interest rate percent at .35% per month.
set present value equal to 0
set future value equal to 0
keep the monthly payments at $352.0923651.
these should be negative as shown by the calculator.
have the calculator tell you what the future value should be.
the calculator will tell you that the future value of those 3 payments is equal to $1059.978378.
multiply that by (1 + .35/100) and the calculator will tell you that the future value of that will be $1063.688303.
add that to the fourth monthly payment of $352.0923651 and the total monthly payment at the end of the fourth month will be $1415.780668.
bottom line:
you need to pay the bank your fourth monthly payment of $352.0923651 plus $1063.688303 to make up for missing the first 3 payments.
the following excel printout shows the remaining balance after 360 monthly payments of $352.09.
the following excel printout shows the remaining balance after 357 monthly payments of $352.09 plus the additional payment of $1063.69 in the fourth month.
the payments in the first 3 months are zero.
as you can see, the remaining balance at the end of 360 months in both cases is $0.00 as it should be.
if you needed to do this manually, the formulas you would need are in the following reference.
https://www.algebra.com/algebra/homework/Finance/THEO-2016-04-29.lesson#notes
the formulas you would need are:
ANNUITY FOR A PRESENT AMOUNT WITH END OF TIME PERIOD PAYMENTS
a = (p*r)/(1-(1/(1+r)^n))
a is the annuity.
p is the present amount.
r is the interest rate per time period.
n is the number of time periods.
and:
FUTURE VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS
f = (a*((1+r)^n-1))/r
f is the future value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods
you would use your calculator and enter the values into the formula, keeping all the parentheses as shown.
for example:
you would start with ANNUITY FOR A PRESENT AMOUNT WITH END OF TIME PERIOD PAYMENTS
the formula is a = (p*r)/(1-(1/(1+r)^n))
p = 72000
r = .0035
n = 360
you would solve for a to get a (72000*.0035)/(1-(1/(1+.0035)^360)) = 352.0923651
you would then use FUTURE VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS
the formula is f = (a*((1+r)^n-1))/r
a = 352.0923651
r = .0035
n = 3
you would solve for f to get f = (352.0923651*((1+.0035)^3-1))/.0035 to get f = 1059.978378.
you would then multiply that by 1.0035 to get 1063.688303.
that's the additional amount you need to pay at the end of the fourth month along with your regular monthly payment of 352.0923651.
the total payment at the end of the fourth month is equal to 1063.688301 + 352.0923651 = 1415.780666.
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your last question is:
he pays the loan for 20 years and, at the end of the 20 year period, he wants to pay off the remaining balance in one lump sum.
if you use your financial calculator, you would do the following:
number of time periods is 360 minus 10*12 = 240 to get 120 remaining payments to be made.
interest rate percent is .35% per month.
present value is 72000
payment each month is -352.0923651.
future value is 0.
you would then calculate for present value of the loan.
you would get $34,451.85545
that's the remaining value of the loan that needs to be paid at the end of 20 years.
if you were to do this manually, you would use the following formula.
PRESENT VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS
p = (a*(1-1/(1+r)^n))/r
p is the present value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods.
you want to know how much you still owe after paying for 240 months.
after 240 months, the number of months remaining is 120.
you would use the PRESENT VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS formula, which is p = (a*(1-1/(1+r)^n))/r
a = 352.0923651
r= .0035
n = 120
you would solve for p = (352.0923651*(1-1/(1+.0035)^120))/.0035 = $34,451.85545
if you look at the excel spreadsheet, it will show you that the remaining balance at the end of 240 months is equal to $34,451.86 as shown below:
if you don't have a financial calculator, you can use one online instead.
the one i use can be found at https://arachnoid.com/finance/
first i used this to calculate the monthly payment as shown below:
the monthly payment is shown as -352.09
then i used it to calculate the future value of 3 months of those monthly payments as shown below:
the future value of the monthly payments is shown as 1059.97
then i multiplied that amount by 1.0035 to get the additional money you needed to pay in the fourth month.
that amount was $1063.88
then i used it to calculate the present value of the payments for the remaining 120 months of the loan, instead of 360 as shown below:
the present value of the payments was shown as 34,451.62
using this online calculator, you will get some small discrepancies between what it tells you and what your own financial calculator might tell you and what the manual formulas might tell you, but the discrepancies are pretty small and, generally speaking, good enough.
depending on how strict your professors are, you would probably want to use a financial calculator, like the texas instruments BA II Plus, or the manual formulas.
this way you get a more exact result that, when rounded, should be the same as your professor might be looking for.
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