Natalie has just one the local lottery of $15,000. The organisers of the lottery offer to pay her in one of the 5 ways below. Natalie can earn 7% interest per annum, compounded monthly on any deposits she makes, which option should she choose?[Fill in the blank]

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BlackTom AI · Accepted Answer

Question restatement: Natalie can choose among five payment options after winning $15,000, with an opportunity to earn 7% interest per year compounded monthly. We evaluate the present value of each option using i = 0.07/12 per month (monthly compounding).

Option A: $15,000 as a once off payment today.
- This is a payment received today, so its present value is simply $15,000. There is no discounting needed because the money is already in hand right now.

Option B: 10 payments of $1,100 at the end of each of the next 10 months.
- The present value of an ordinary annuity with payment P = 1100, monthly rate i = 0.07/12, and n = 10 is PV = P * [1 - (1 + i)^(-n)] / i.
- Compute i ≈ 0.0058333. (1 + i)^(-10) ≈ 1 / (1.0058333^10) ≈ 1 / 1.0599 ≈ 0.9439.
- Therefore PV ≈ 1100 * (1 - 0.9439) / 0.0058333 ≈ 1100 * 0.0561 / 0.0058333 ≈ 1100 * 9.62 ≈ $10,580.
- This is less than $15,000, so this option has a smaller value today.

Option C: A perpetuity of $60 each month.
- PV of a monthly perpetuity = payment / monthly rate = 60 / 0.0058333.
- PV ≈ 60 / 0.0058333 ≈ $10,286.
- This is also less than $15,000.

Option D: $16,700 in exactly 4 years.
- Present value = 16700 / (1 + i)^(48), since 4 years = 48 months.
- (1 + i)^(48) ≈ (1.0058333)^(48) ≈ about 1.32.
- PV ≈ 16700 / 1.32 ≈ $12,642.
- Still less than $15,000 today.

Option E: $15,100 at the end of year (t = 1).
- We must use the annual effective rate corresponding to 7% compounded monthly: (1 + 0.07/12)^12 − 1 ≈ (1.0058333)^{12} − 1 ≈ 0.0723, i.e., about 7.23% effective annual rate.
- PV = 15100 / (1 + 0.0723) ≈ 15100 / 1.0723 ≈ $14,100–$14,090 (approximate depending on rounding).
- This is still less than $15,000 today.

Summary of comparisons:
- A provides PV = $15,000 (the actual amount today).
- B, C, D, and E all have present values well below $15,000 when discounted at 7% nominal monthly-compounded interest (monthly rate 0.5833%), either due to periodic payments being in the future, higher total nominal amounts but discounted over time, or using a different effective annual rate.

Conclusion (inferred from present-value calculations): The option whose present value matches the immediate amount best satisfies the comparison, meaning the option that gives $15,000 today has the highest present value among the given choices.

Natalie has just one the local lottery of $15,000. The organisers of the lottery offer to pay her in one of the 5 ways below. Natalie can earn 7% interest per annum, compounded monthly on any deposits she makes, which option should she choose?[Fill in the blank]

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