A 20 year loan requires semi-annual payments of $1333.28 including interest at 10.75% compounded semi annually. what is the original amount of the loan and what will be the balance of the loan 8.5 years later (just after the scheduled payment?)
PV = 1333.28(1 - 1.05375^-40)/.05375
= $21750.00
Balance after 8.5 year
= 21750.00(1.05375)^17 - 1333.28(1.05375^17 - 1/.05375
= $17365.12
check my arithmetic
To find the original amount of the loan, you can use the formula for the present value of an ordinary annuity. The formula is:
PV = P * (1 - (1 + r)^(-n)) / r
Where:
PV is the present value or original amount of the loan
P is the periodic payment
r is the interest rate per period
n is the total number of periods
In this case, the periodic payment is $1333.28, the interest rate is 10.75% compounded semiannually, and the loan is for 20 years or 40 semiannual periods.
First, let's calculate the interest rate per semiannual period:
r = 10.75% / 2
r = 0.1075 / 2
r = 0.05375
Next, substitute the values into the formula:
PV = $1333.28 * (1 - (1 + 0.05375)^(-40)) / 0.05375
Now, calculate the present value:
PV = $1333.28 * (1 - (1.05375)^(-40)) / 0.05375
PV = $1333.28 * (1 - 0.170584) / 0.05375
PV = $1333.28 * 0.829416 / 0.05375
PV = $20774.76
Therefore, the original amount of the loan is $20,774.76.
To find the balance of the loan 8.5 years later, we need to calculate the remaining balance after 34 semiannual payments. We can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value or remaining balance of the loan after 8.5 years
P is the periodic payment
r is the interest rate per period
n is the number of periods remaining (40 - 34 = 6 semiannual periods)
Substituting the values into the formula:
FV = $1333.28 * ((1 + 0.05375)^6 - 1) / 0.05375
Now, calculate the future value:
FV = $1333.28 * ((1 + 0.05375)^6 - 1) / 0.05375
FV = $1333.28 * (1.353401 - 1) / 0.05375
FV = $1333.28 * 0.353401 / 0.05375
FV = $8,777.05
Therefore, the balance of the loan 8.5 years later, just after the scheduled payment, would be $8,777.05.