What is the annualized cost of the debt for a sinking fund that has semi-annual deposits and it is set up to pay off a debt of $20 000 over a period of 5 years? The sinking fund earns 4.6% compounded annually and the loan rate on the debt is 9.2% compounded quarterly.

Question

Quizplus · Accepted Answer

Step 1: Calculate the semi-annual deposit amount
The total number of semi-annual periods is 2*5 = 10
The sinking fund needs to accumulate $20 000, so the semi-annual deposit amount is:
PMT = $20 000 / 10 = $2 000

Step 2: Calculate the future value of the sinking fund
The sinking fund earns 4.6% compounded annually, which is equivalent to 2.3% compounded semi-annually.
Using the formula for future value of an annuity, we have:
FV = PMT * ((1 + r)^n - 1) / r
where r is the semi-annual interest rate and n is the number of semi-annual periods
FV = $2 000 * ((1 + 0.023)^10 - 1) / 0.023
FV = $22 877.44

Step 3: Calculate the annualized cost of the debt
The loan rate on the debt is 9.2% compounded quarterly, which is equivalent to 2.3% compounded quarterly.
Using the formula for present value of an annuity, we have:
PV = PMT * (1 - (1 + r)^-n) / r
where r is the quarterly interest rate and n is the number of quarterly periods
n = 4 * 5 = 20
PV = $2 000 * (1 - (1 + 0.023/4)^-20) / (0.023/4)
PV = $16 430.92
The total cost of the debt is $16 430.92.
The annualized cost of the debt is the rate that would make the present value of the payments equal to the total cost of the debt. Using a financial calculator or spreadsheet, we find that the annualized cost is 11.9053%.
Therefore, the annualized cost of the debt is $2 000 * 2 * 11.9053% = $5447.52.

What Is the Annualized Cost of the Debt for a Sinking