A company requires the amount of $645,000 in 15 years to retire a bond issue. Assume they can earn 4 1/2 % interest compounded daily. What amount would they have to pay quarterly to be able to retire this debt in 15 years?
To calculate the quarterly payment amount required to retire the bond issue in 15 years, we can use the formula for the future value of an ordinary annuity:
PV = PMT × [(1 - (1 + r/q)^(-n*q)) / (r/q)]
Where:
PV = Present value (amount required to retire the bond)
PMT = Payment amount
r = Annual interest rate (as a decimal)
q = Number of compounding periods per year
n = Number of years
Given:
PV = $645,000
r = 4.5% = 0.045 (as a decimal)
q = 4 (quarterly compounding)
n = 15
Substituting these values into the formula:
$645,000 = PMT × [(1 - (1 + 0.045/4)^(-15*4)) / (0.045/4)]
Now, let's solve for PMT step by step.
Step 1: Calculate the fraction inside the square brackets:
(1 - (1 + 0.045/4)^(-15*4)) / (0.045/4)
= (1 - (1.01125)^(-60)) / 0.01125
Step 2: Evaluate the exponent:
(-60)
Step 3: Calculate the value inside the square brackets:
= (1 - 0.314511) / 0.01125
= 0.685489 / 0.01125
= 61.01688888888889
Step 4: Solve for PMT:
$645,000 = PMT × 61.01688888888889
Step 5: Divide both sides by 61.01688888888889:
PMT = $645,000 / 61.01688888888889
Calculating the final value:
PMT ≈ $10,578.46
Therefore, the company would need to pay approximately $10,578.46 quarterly to retire the debt in 15 years.
To calculate the quarterly payment required to retire the debt in 15 years, we can use the formula for the present value of an annuity.
The formula for the present value of an annuity is:
PV = PMT × [1 - (1 + r/n)^(-nt)] / (r/n)
Where:
PV = Present Value (the amount needed to retire the bond)
PMT = Periodic payment (the quarterly payment)
r = Annual interest rate (converted to decimal)
n = Number of compounding periods per year
t = Number of years
Given values:
PV = $645,000
r = 4.5% = 0.045 (as a decimal)
n = 365 (since interest is compounded daily)
t = 15 years
Substituting these values into the formula, we get:
645,000 = PMT × [1 - (1 + 0.045/365)^(-365*15)] / (0.045/365)
Now, let's solve this equation to find the quarterly payment (PMT).
First, simplify the expression inside the brackets:
645,000 = PMT × (1 - 1.000123^(5475)) / 0.000123
Next, calculate (1.000123^(5475)):
645,000 = PMT × (1 - 3.363163095) / 0.000123
Now, subtract 3.363163095 from 1, and divide the result by 0.000123:
645,000 = PMT × ( -2.363163095) / 0.000123
Simplifying further:
645,000 = PMT × (-19,209.71738)
Dividing both sides of the equation by (-19,209.71738), we can isolate PMT:
PMT = 645,000 / (-19,209.71738)
PMT ≈ $-33.60 (rounded to two decimal places)
The quarterly payment would be approximately -$33.60. The negative sign indicates that it is an outgoing payment (outflow of cash) to retire the bond.