Final answer:
To find out how much money the state must have in hand to set up the payments for such a lottery prize, the present value of an annuity formula should be used, factoring in the monthly payments, interest rate, and duration of the annuity.
Step-by-step explanation:
To determine how much money the state needs in hand to set up the payments for a $2.4 million lottery with $10,000 monthly payments over 20 years at an interest rate of 6.3% compounded monthly, we need to find the present value of an annuity. The formula for the present value of an annuity is given by:
PV = PMT × [(1 - (1 + r)^{-n}) / r]
Where:
- PV is the present value of the annuity,
- PMT is the monthly payment amount,
- r is the monthly interest rate,
- n is the total number of payments.
In this case:
- PMT = $10,000,
- r = 6.3% per year, or 0.063 / 12 per month,
- n = 20 years × 12 months/year = 240 months.
Thus, the present value of the annuity (amount the state needs to have) is calculated as follows:
PV = $10,000 × [(1 - (1 + 0.063 / 12)^{-240}) / (0.063 / 12)]
We first convert the annual interest rate to monthly by dividing by 12 and then apply the formula to calculate PV.
By doing the calculation, the state will find out the lump sum amount needed to put into an account today to meet the payout requirements.