A major lottery advertises that it pays the winner $10 million. However this prize money is paid at the rate of $500,000 each year (with the first payment being immediate) for a total of 20 payments. What is the present value of this prize at 10% interest compounded annually?
Report your answer in $millions, rounded to two decimal places. So, for example, if you compute the answer to be 5.7124 million dollars then you should submit an answer of 5.71.
To find the present value of the prize money, we need to calculate the value of each individual payment and then sum them up.
The present value of a future payment formula is:
PV = Payment / (1 + r)^n
Where PV is the present value, Payment is the annual payment, r is the interest rate, and n is the number of periods.
In this case, the annual payment is $500,000, the interest rate is 10% (or 0.10), and there are 20 periods.
Let's calculate the present value of each payment:
PV1 = $500,000 / (1 + 0.10)^1
PV2 = $500,000 / (1 + 0.10)^2
PV3 = $500,000 / (1 + 0.10)^3
...
PV20 = $500,000 / (1 + 0.10)^20
Now, let's sum up all the present values to find the total present value:
PV_total = PV1 + PV2 + PV3 + ... + PV20
Using a financial calculator or a spreadsheet, we can calculate the present value of each payment and then sum them up.
PV_total = $500,000 / (1 + 0.10)^1 + $500,000 / (1 + 0.10)^2 + $500,000 / (1 + 0.10)^3 + ... + $500,000 / (1 + 0.10)^20
Calculating this sum, we get:
PV_total ≈ $5.75 million (rounded to two decimal places)
Therefore, the present value of the prize money is approximately $5.75 million.