If money is worth 12% compounded quarterly, determine the single payment at the end of the fourth year that will replace a uniform annual series of P2,500.00 for 14 years.
To solve this problem, we need to find the present value of the uniform annual series of P2,500.00 for 14 years, and then calculate the equivalent single payment at the end of the fourth year at the given interest rate of 12% compounded quarterly.
Step 1: Find the present value of the uniform annual series.
The present value of a uniform annual series can be calculated using the formula for the present value of an annuity:
PV = C * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present Value
C = Cash flow (annual payment)
r = Interest rate per compounding period
n = Number of compounding periods
In this case, the cash flow is P2,500.00, and the number of compounding periods is 14 years. The annual interest rate is 12%, so the interest rate per compounding period would be 12% / 4 = 3% (since it is compounded quarterly).
Plugging these values into the formula:
PV = 2500 * [(1 - (1 + 0.03)^(-14)) / 0.03]
Using a calculator, evaluate the expression inside the brackets first [(1 - (1 + 0.03)^(-14)) / 0.03], and then multiply it by 2500 to get the present value.
Step 2: Calculate the equivalent single payment at the end of the fourth year.
To find the equivalent single payment at the end of the fourth year, we need to compound the present value calculated in step 1 to the end of the fourth year.
Using the compound interest formula:
A = PV * (1 + r)^n
Where:
A = Future value (equivalent single payment)
PV = Present Value (calculated in step 1)
r = Interest rate per compounding period (12% compounded quarterly would be 12% / 4 = 3%)
n = Number of compounding periods (4 years)
Plug in the values and calculate A.
By following these steps, you can determine the single payment at the end of the fourth year that will replace the uniform annual series.