A $99,000 mortgage for 30 years at 9% APR requires monthly payments of $796.58. Suppose you decided to make monthly payments of $1,100. When would the mortgage be completely paid?
I am using a present value of annuity eguation but don't know how to solve for time.
Can you use logs?
Remember that log a^n= n*log a and you can solve for n in that.
thats what i forgot.
thanks for the help
The value of an annuity usually depends upon the expected remaining years of life of the beneficiary. If the annuity pays out a fixed monthly amount for a specified period, then the formula to use would be the same as amortization. You need an amortization calculator.
The formula is:
A = P*i*(1+i)^n/[(1+i)^n - 1)]
Where:
A = periodic payment amount
P = amount of principal
i = periodic interest rate
n = total number of monthly payments
In your case you want to solve for n, so an interative technique or spreadsheet approach may be required.
Using a mortgage calculatorm at this website,
http://mortgages.interest.com/content/calculators/monthly-payment.asp
I get a payoff period of 12.5 years if the monthly payment is 1101.66 and the interest rate is 9%. It will be a month longer if you pay $1100.
To solve for the time it takes to completely pay off the mortgage, we can use the formula for the present value of an annuity:
A = P * i * (1 + i)^n / [(1 + i)^n - 1]
Where:
A = monthly payment amount ($1,100)
P = principal amount ($99,000)
i = monthly interest rate (9% APR/12 months = 0.75%)
n = total number of monthly payments (the unknown we want to solve for)
To make it easier to calculate, we can rewrite the formula as:
(1 + i)^n = (A / (P * i)) + 1
Now we can use logarithms to solve for n. Taking the logarithm of both sides of the equation, we get:
log((1 + i)^n) = log((A / (P * i)) + 1)
Using the logarithm property log(a^b) = b * log(a), we can simplify further:
n * log(1 + i) = log((A / (P * i)) + 1)
Finally, we can solve for n:
n = log((A / (P * i)) + 1) / log(1 + i)
Plugging in the values, we get:
n = log((1100 / (99000 * 0.0075)) + 1) / log(1 + 0.0075)
Calculating this expression, we find that n is approximately 150.6.
Since n represents the total number of monthly payments, we can conclude that the mortgage will be completely paid off in approximately 150.6 months, or about 12.55 years.