What minimum amount will have to be deducted today to a fund earning 8.7% compounded quarterly, if the first quarterly payment of $6000 in perpetuity is to occur three months from now.
Do not round intermediate calculations, just round the final answer to 2 decimal places
To calculate the minimum amount that needs to be deducted today, we need to find the present value of perpetuity payments three months from now.
The formula to calculate the present value of a perpetuity is:
PV = PMT / (1 + r)^n
Where:
PV = Present Value
PMT = Payment amount
r = Interest rate per period
n = Number of periods
In this case, the payment amount (PMT) is $6000, the interest rate per period (r) is 8.7% compounded quarterly, and the number of periods (n) is three months.
First, we need to calculate the interest rate per period (r). Since the interest is compounded quarterly, we divide the annual interest rate by 4:
r = 8.7% / 4 = 0.087 / 4 = 0.02175
Next, we need to calculate the number of periods (n) in terms of quarters. Since the perpetuity payment is three months from now, we consider it as occurring in the first quarter:
n = 1
Now we can calculate the present value (PV) by substituting the values into the formula:
PV = $6000 / (1 + 0.02175)^1
Calculating the value inside the parentheses:
(1 + 0.02175)^1 = 1.02175
PV = $6000 / 1.02175
Calculating the present value:
PV ≈ $5869.11 (rounded to 2 decimal places)
Therefore, the minimum amount that needs to be deducted today to a fund earning 8.7% compounded quarterly, for the first quarterly payment of $6000 in perpetuity to occur three months from now, is approximately $5869.11 (rounded to 2 decimal places).