Parents have set up a sinking fund in order to have $120,000 in 15 years for their children's education. How much should be paid semiannually into an account paying 6.8% compounded semiannually?
PMT = FV ____i_____
(1 + i)^n – 1
To calculate the amount that should be paid semiannually into the sinking fund, we can use the formula for the future value of an annuity:
FV = PMT * [(1 + i)^n - 1] / i
Where:
FV = Future value of the sinking fund ($120,000 in this case)
PMT = Payment made semiannually
i = Interest rate per compounding period (6.8% divided by 2, since it is compounded semiannually)
n = Number of compounding periods (15 years multiplied by 2, since it is compounded semiannually)
Let's substitute the given values into the formula:
$120,000 = PMT * [(1 + 0.068/2)^(15*2) - 1] / (0.068/2)
Simplifying the equation:
$120,000 = PMT * [(1 + 0.034)^30 - 1] / 0.034
Now, let's rearrange the equation to solve for PMT:
PMT = $120,000 * (0.034 / [(1 + 0.034)^30 - 1])
Using a calculator, evaluate the expression in the brackets [(1 + 0.034)^30 - 1]:
PMT = $120,000 * (0.034 / (1.034^30 - 1))
PMT ≈ $120,000 * (0.034 / 1.758901)
PMT ≈ $2,334.32
Therefore, the parents should pay approximately $2,334.32 semiannually into the account.
To determine the amount that should be paid semiannually into the sinking fund, we can use the sinking fund formula.
The formula for calculating the required payment (PMT) into a sinking fund is as follows:
PMT = FV [/ {(1 + i) ^ n - 1} / i]
Where:
PMT = Payment amount
FV = Future value (the desired amount in the sinking fund)
i = Interest rate per compounding period (in this case, semiannually)
n = Number of compounding periods
In this case, the future value (FV) is $120,000, the interest rate (i) is 6.8% compounded semiannually, and the number of periods (n) is 15 years, which corresponds to 30 semiannual periods.
Plugging these values into the formula, we have:
PMT = $120,000 [/ {(1 + 0.068/2) ^ (15 * 2) - 1} / (0.068/2)]
Now let's solve it step by step:
First, let's calculate the denominator:
Denominator = {(1 + 0.034) ^ 30 - 1}
Next, compute the numerator:
Numerator = $120,000
Finally, divide the numerator by the denominator, and then multiply by the interest rate:
PMT = $120,000 / {(1 + 0.034) ^ 30 - 1} / 0.034
By evaluating this expression, we can find the semiannual payment that should be made into the account.
You have the right formula typically shown in the form or
Ri = Sn/[(1+i)^n - 1]
Sn = 120,000
n = 15x2 = 30
i = .068/2 = .034
Make the keys dance.