Maisy has won the lottery and will receive $50,000 at the end of each six-month period over the next five years. Maisy wants to find out what this win is equivalent to in today’s money, given that the interest rate is 4.75% per year. Her calculations indicate that her lottery win amount to $440,443 today (rounded to the closest dollar).[Fill in the blank]判断题
A
True
B
False

Question

Maisy has won the lottery and will receive $50,000 at the end of each six-month period over the next five years. Maisy wants to find out what this win is equivalent to in today’s money, given that the interest rate is 4.75% per year. Her calculations indicate that her lottery win amount to $440,443 today (rounded to the closest dollar).[Fill in the blank]判断题

A

True

B

False

BlackTom AI · Accepted Answer

Present value of an ordinary annuity uses the factor PV = PMT × [1 − (1 + i)^−n] / i. Here payments are $50,000 at the end of each six‑month period for 5 years, so PMT = 50,000, number of periods n = 5 × 2 = 10, and the periodic interest rate is i = 0.0475 / 2 = 0.02375 (2.375% per half‑year).

Step‑by‑step:
1. Compute the periodic rate: i = 4.75% / 2 = 2.375% = 0.02375.
2. Compute (1 + i)^n: (1.02375)^10 ≈ 1.264559148 (calculator value shown to 9 decimals).
3. Compute the discount factor (1 + i)^−n = 1 / 1.264559148 ≈ 0.790987 (rounded at this stage).
4. Compute the annuity factor numerator: 1 − (1 + i)^−n ≈ 1 − 0.790987 = 0.209013.
5. Divide by i: 0.209013 / 0.02375 ≈ 8.808859 (annuity present value factor).
6. Multiply by PMT: PV = 50,000 × 8.808859 ≈ 440,442.95, which rounds to $440,443.

Why this is correct: the formula used is the standard present‑value formula for an ordinary annuity (payments at the end of each period). Each intermediate quantity (periodic rate, growth factor, discount factor, annuity factor) was computed with the correct n and i for semiannual compounding, and the final multiplication yields the stated present value rounded to the nearest dollar.

Why the other option (False) is incorrect: any alternative that claims the present value is not $440,443 would have to use a different interest conversion (e.g., wrong periodic rate or wrong number of periods) or an incorrect formula (e.g., annuity due vs. ordinary). Using the correct ordinary‑annuity formula with semiannual compounding gives $440,443.

Conclusion: the computed present value is $440,443 (rounded), so the statement is True.

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