the interest rate is 4.4% compounded monthly.
your payment is going to be equal to x.
the loan is paid off when the present value of the payments is equal to the present value of the loan.
the present value of the loan is 16,000.
the annual interest rate is 4.4%.
divide that by 100 to get an annual interest rate of .044.
divide that by 12 to get a monthly interest rate of .044/12.
the first payment is 2 years out which is 24 months out.
the second payment is 6 years out which is 72 months out.
the present value of the first payment is therefore x / (1 + .044/12)^24.
the present value of the second payment is therefore x / (1 + .044/12)^72.
for now, we'll let (1 + .044/12)^24 be equal to a, and we'll let (1 + .044/12)^72 be equal to b so we don't have to write them down so many times.
the present value of the payments are therefore x / a and x / b.
the present value must be equal to 16,000, so we get 16,000 = x/a + x/b.
multiply both sides of this equation by a * b and we get:
16,000 * a * b = x * b + x * a
factor out the x to get 16,000 * a * b = x * (b + a)
divide both sides of this equation by (b + a) to get:
(16,000 * a * b) / (b + a) = x
since a = (1 + .044/12)^24 and b = (1 + .044/12)^72, we can solve for x to get:
x = 9499.762157.
that's your payment in 24 months and in 72 months.
the present value of the first payment is 9499.762157 / (1 + .044/12)^24 = 8700.910734.
the present value of the second payment is 9499.762157 / (1 + .044/12)^72 = 7299.089266.
the sum of the present value of the payments is equal to 16000 which means the loan is paid off.
this can be seen in more detail in the following excel printout.
the coments after the first payment is made should say the calculations for timer period 24 are, rather than the calculations for time period 4 are.
that was a typo.
the general form of the equation for future value is:
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time period.
in your problem, the interest rate was 4.4% per year.
that was divided by 100 to get an interest rate of .044 per year.
that was divided by 12 to get an interest rate of .044/12 per month.
since it was a repeating decimal, i left it as is rather than showing .044/12 as .036666666666666666666.......
to find the present value, you would use the same formula solved for p rather than f.
that would be p = f / (1 + r) ^ n
this is the formula used to bet the present value of the payments.
the payments were x.
the present value of the first payment was x / (1 + .044/12)^24.
the present value of the second payment was x / (1 + .044/12)^72.