Find the present value of the annuity necessary to fund the withdrawal of $600 per month for 10 years, if the annuity earns 2% per year and if there is to be $10,000 to be left in the annuity at the end of the 10 years. (Assume end-of-period withdrawal and compounding at the same intervals as withdrawal. Round your answer to the nearest ten cents.)
To find the present value of the annuity necessary to fund the withdrawal of $600 per month for 10 years, we can use the formula for the present value of an ordinary annuity:
PV = PMT * ((1 - (1 + r)^(-n)) / r),
where:
PV is the present value of the annuity,
PMT is the withdrawal amount per period (in this case, $600),
r is the interest rate per period (2% per year, or 0.02/12 per month),
n is the total number of compounding periods (in this case, 10 years, or 120 months).
First, calculate the present value of the annuity:
PV = 600 * ((1 - (1 + 0.02/12)^(-120)) / (0.02/12)).
To simplify the calculation, let's break it down step by step:
1. Calculate the fraction within the parentheses: (1 - (1 + 0.02/12)^(-120)).
- Add 1 to the monthly interest rate: 1 + 0.02/12 = 1.00167.
- Raise this value to the power of the negative number of months: (1.00167)^(-120).
- Subtract this value from 1: 1 - (1.00167)^(-120) = 0.71449 (rounded to 5 decimal places).
2. Calculate the fraction outside the parentheses: (0.02/12).
- Divide the annual interest rate by 12: 0.02/12 = 0.00167 (rounded to 5 decimal places).
3. Calculate the present value by dividing the two fractions: PV = 600 * (0.71449 / 0.00167).
- PV = $256,563.47 (rounded to the nearest cent).
Therefore, the present value of the annuity necessary to fund the withdrawal of $600 per month for 10 years, with an interest rate of 2% per year and $10,000 to be left at the end, is approximately $256,563.47.