Sarah just won a $3,000,000.00 lottery. According to the lottery they will pay her a lump sum of $500,000.00 on October 1, 2012 and the balance in equal annual installments for 10 years.

Assuming that you are the director of the lottery, how much must you have in the account to pay Sarah her initial payment and the 10 remaining payments. Also assume that you have a discount rate of 6%

Are we to assume they going to pay out the 3 Million dollar win with a payment of $500,000 now followed by 10 annual payments of 250,000 ?

(Then she did not really win $3 million, since "time is worth money" )
According to the wording it seems like that is the assumption, so
present value = 500,000 + 250000(1 - 1.06^-10)/.06
= 2,340,021.76

A more realistic question would have been, to find each of the 10 annual payments plus the present payment of 500,00 which would be equivalent to 3 million now.

65656

To calculate the amount you must have in the account to pay Sarah her initial payment and the remaining payments, you need to calculate the present value of the future cash flows.

First, let's calculate the present value of the initial payment:

PV (Present Value) = FV (Future Value) / (1 + r)^n

Where:
FV = $500,000.00 (future value of the initial payment)
r = 6% (discount rate)
n = number of periods until the initial payment is made (since October 1, 2012, is in the past, n will be the number of periods from today until October 1, 2012)

To calculate the number of periods, you can use a date calculator or count the number of years from today (September 2021) to October 1, 2012. Let’s assume it is exactly 9 years.

n = 9

Calculating the present value of the initial payment:

PV = $500,000.00 / (1 + 0.06)^9
PV = $500,000.00 / (1.06)^9
PV ≈ $314,047.54

Next, let's calculate the present value of the annual payments for the remaining 10 years.

Since these are equal annual installments, you can use the Present Value of an Ordinary Annuity (PVA) formula:

PVA = A * [1 - (1 + r)^(-n)] / r

Where:
A = annual payment
r = 6% (discount rate)
n = number of periods (remaining 10 years)

To calculate A, you need to determine the annual payment that will be paid over the 10-year period. The remaining balance after the initial payment is the total prize amount minus the initial payment:

Remaining balance = $3,000,000.00 - $500,000.00
Remaining balance = $2,500,000.00

To calculate the annual payment:

A = remaining balance / (1 - (1 + r)^(-n)) / r

A = $2,500,000.00 / (1 - (1 + 0.06)^(-10)) / 0.06
A ≈ $351,152.82

Now, let's calculate the present value of the annual payments:

PVA = $351,152.82 * [1 - (1 + 0.06)^(-10)] / 0.06
PVA ≈ $2,511,882.75

Finally, to determine the total amount you must have in the account to pay Sarah, you add the present value of the initial payment to the present value of the annual payments:

Total amount = PV of initial payment + PV of annual payments
Total amount = $314,047.54 + $2,511,882.75
Total amount ≈ $2,825,930.29

Therefore, as the director of the lottery, you must have approximately $2,825,930.29 in the account to pay Sarah her initial payment and the remaining payments.