To calculate the present worth of the cost for a 3-year time period at a 20% MARR compounded quarterly, you need to calculate the future values of the cost increase and then discount them to the present. First, calculate the future values for each quarter using the future value formula. Then, discount each future value to the present using the present value formula. Finally, sum up all the present values to find the present worth of the cost.
To calculate the present worth of the cost for a 3-year time period, you will need to calculate the future values of the cost increase and then discount them to the present using the MARR of 20% compounded quarterly. Here are the steps to follow:
First, calculate the future value of the cost increase for each quarter. Since the cost increases by $300 every 3 months, this means it will increase by $300 for each quarter for a total of 12 quarters in 3 years. The future value for each quarter can be calculated using the future value formula:
Future Value = Present Value * (1 + Interest Rate)^n
where:
- Present Value = $300
- Interest Rate = 20% / 4 (since it is compounded quarterly)
- n = number of quarters = 12
Calculating the future value for each quarter:
Future Value (Quarter 1) = $300 * (1 + 0.20/4)^1 = $309.54
Future Value (Quarter 2) = $300 * (1 + 0.20/4)^2 = $319.47
...
Future Value (Quarter 12) = $300 * (1 + 0.20/4)^12 = $448.79
Next, discount each future value to the present using the present value formula:
Present Value = Future Value / (1 + Interest Rate)^n
where:
- Future Value = calculated from step 1
- Interest Rate = 20% / 4
- n = number of quarters to discount = 0 for the first quarter and increases by 1 for each subsequent quarter
Calculating the present value for each quarter:
Present Value (Quarter 1) = $309.54 / (1 + 0.20/4)^0 = $309.54
Present Value (Quarter 2) = $319.47 / (1 + 0.20/4)^1 = $313.14
...
Present Value (Quarter 12) = $448.79 / (1 + 0.20/4)^11 = $239.39
Finally, sum up all the present values to get the present worth of the cost for the 3-year time period:
Present Worth = Present Value (Quarter 1) + Present Value (Quarter 2) + ... + Present Value (Quarter 12)
Calculating the present worth:
Present Worth = $309.54 + $313.14 + ... + $239.39 = $3,045.35