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?
$30,000
End of 8 years
12% interest
8*2= 16 periods
12/2= 6%
Go to the table in your math procedures book and look under sinking fund table (Practical Business Math Procedures Jeffery Slater/ Sharon M. Whittry 12th edition Page 12.)
table =.0390
$30,000 * .0390= $1170.00
let P be the payment
30000 = P(1.06^16 - 1)/.06
solve for P
that should have been
30000 = P(1 - 1.06^-16)/.06
To calculate the periodic payment Jeff must make into the sinking fund, we can use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value (loan amount)
P = Periodic Payment
r = Interest rate per period
n = Number of periods
In this case, the loan amount (FV) is $30,000, the interest rate (r) is 12% (0.12) compounded semiannually, and the loan term (n) is 8 years. Since the loan term is given in years, but the compounding is semiannually, we need to adjust the number of periods accordingly.
Number of compounding periods per year = 2 (semiannually)
Number of compounding periods in total = Number of years x Number of compounding periods per year
So in this case, the number of compounding periods in total is 8 years x 2 = 16 periods.
Let's substitute the values into the formula:
30,000 = P * ((1 + 0.12/2)^16 - 1) / (0.12 / 2)
Now, solve the equation to find the periodic payment (P):
P = 30,000 * (0.12/2) / (((1 + 0.12 / 2)^16) - 1)
Using a calculator or spreadsheet, evaluate the expression to find the value of P. The result will be the amount Jeff must pay into the sinking fund each period of time to repay the loan amount at the end of 8 years.