Questions: Find the cash value of the following lottery jackpot. Yearly jackpot payments begin immediately. Assume the lottery can invest at the given interest rate. 14,000,000 jackpot, 6% interest, 20 annual payments (Round to the nearest dollar as needed.)

Find the cash value of the following lottery jackpot. Yearly jackpot payments begin immediately. Assume the lottery can invest at the given interest rate. 14,000,000 jackpot, 6% interest, 20 annual payments (Round to the nearest dollar as needed.)
Transcript text: Find the cash value of the following lottery jackpot. Yearly jackpot payments begin immediately. Assume the lottery can invest at the given interest rate. $\$ 14,000,000$ jackpot, $6 \%$ interest, 20 annual payments (Round to the nearest dollar as needed.)
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Solution

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Solution Steps

Step 1: Calculate the Annual Payment (P)

To find the annual payment, divide the total jackpot amount by the number of payments. Given a jackpot amount of $14000000 to be paid out over 20 years, the annual payment \(P\) is calculated as: \[ P = \frac{FV}{n} = \frac{14000000}{20} = 700000 \]

Step 2: Calculate the Present Value (PV)

Using the present value of an annuity formula, we can calculate the present value of the lottery winnings. The formula is: \[ PV = P \times \left(1 - (1 + r)^{-n}\right) / r \] Substituting the values, we get: \[ PV = 700000 \times \left(1 - (1 + 0)^{-20}\right) / 0 = 8028945 \]

Final Answer:

The present value of the annuity, given a jackpot amount of $14000000, an annual interest rate of 6%, and 20 equal annual payments, is approximately $8028945.

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