Jeff Associates borrowed $30,000. The company plans to set up a sinking fund that will repay the loan at the end of 8 years. Assume a 12 % interest rate compounded semiannually. What must Jeff pay into the fund each period of time?
Let the semi-annual payment be P
30000 = P( 1 - 1.06^-16)/.06
let me know what you got for P
To determine the amount Jeff must pay into the sinking fund each period of time, we can use the formula for the future value of an ordinary annuity. This formula can be used to calculate the amount needed for loan repayment over a specific period of time.
The formula for the future value of an ordinary annuity is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value of the annuity (loan amount)
P = Payment amount made each period
r = Interest rate per period
n = Number of periods
In this case, Jeff plans to repay the loan over 8 years, with a 12% interest rate compounded semiannually.
Since the loan repayment is semiannually, the number of periods will be 8 years * 2 = 16 periods.
Now, we can plug in the values into the formula and solve for P:
30000 = P * ((1 + 0.12/2)^16 - 1) / (0.12/2)
Let's calculate it step by step:
Step 1: Convert the interest rate to a decimal and divide it by the compounding frequency:
Annual interest rate = 12% = 0.12
Interest rate per period = 0.12/2 = 0.06
Step 2: Calculate the future value using the formula:
30000 = P * ((1 + 0.06)^16 - 1) / 0.06
Step 3: Simplify the formula:
30000 = P * (1.06^16 - 1) / 0.06
Step 4: Calculate the value of (1.06^16 - 1):
(1.06^16 - 1) ≈ 1.21816
Step 5: Substitute the value back into the formula:
30000 = P * 1.21816 / 0.06
Step 6: Multiply both sides by 0.06 to isolate P:
0.06 * 30000 = P * 1.21816
Step 7: Divide both sides by 1.21816 to solve for P:
P ≈ (0.06 * 30000) / 1.21816
By evaluating the equation, we find that Jeff must contribute approximately $1,471.28 into the sinking fund each period of time to repay the loan at the end of 8 years.