Question

 ​(Present value of annuity​ payments)  The state​ lottery's million-dollar payout provides for ​$1 million to be paid in 25 installments of ​$40000 per payment. The first ​$40000 payment is made​ immediately, and the 24 remaining ​$40000 payments occur at the end of each of the next 24 years. If 6 percent is the discount​ rate, what is the present value of this stream of cash​ flows? If 12 percent is the discount​ rate, what is the present value of the cash​ flows?

Answer 1

The present value is computed as shown below:

= Future value / (1 + r)ⁿ

So, the present value at 6% discount rate will be as follows:

= $ 40,000 + $ 40,000 / 1.06¹ + $ 40,000 / 1.06² + $ 40,000 / 1.06³ + $ 40,000 / 1.06⁴ + $ 40,000 / 1.06⁵ + $ 40,000 / 1.06⁶ + $ 40,000 / 1.06⁷ + $ 40,000 / 1.06⁸ + $ 40,000 / 1.06⁹ + $ 40,000 / 1.06¹⁰ + $ 40,000 / 1.06¹¹ + $ 40,000 / 1.06¹² + $ 40,000 / 1.06¹³ + $ 40,000 / 1.06¹⁴ + $ 40,000 / 1.06¹⁵ + $ 40,000 / 1.06¹⁶ + $ 40,000 / 1.06¹⁷ + $ 40,000 / 1.06¹⁸ + $ 40,000 / 1.06¹⁹ + $ 40,000 / 1.06²⁰ + $ 40,000 / 1.06²¹ + $ 40,000 / 1.06²² + $ 40,000 / 1.06²³ + $ 40,000 / 1.06²⁴

= $ 542,014.30 Approximately

The present value at 12% discount rate is computed as follows:

= $ 40,000 + $ 40,000 / 1.12¹ + $ 40,000 / 1.12² + $ 40,000 / 1.12³ + $ 40,000 / 1.12⁴ + $ 40,000 / 1.12⁵ + $ 40,000 / 1.12⁶ + $ 40,000 / 1.12⁷ + $ 40,000 / 1.12⁸ + $ 40,000 / 1.12⁹ + $ 40,000 / 1.12¹⁰ + $ 40,000 / 1.12¹¹ + $ 40,000 / 1.12¹² + $ 40,000 / 1.12¹³ + $ 40,000 / 1.12¹⁴ + $ 40,000 / 1.12¹⁵ + $ 40,000 / 1.12¹⁶ + $ 40,000 / 1.12¹⁷ + $ 40,000 / 1.12¹⁸ + $ 40,000 / 1.12¹⁹ + $ 40,000 / 1.12²⁰ + $ 40,000 / 1.12²¹ + $ 40,000 / 1.12²² + $ 40,000 / 1.12²³ + $ 40,000 / 1.12²⁴

= $ 351,372.63 Approximately

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