If you could invest the money at 8%, how much would you have to invest?

Hint: compare your numerical answer with the previous question, would you have to invest more or less than at 6%?

Question

If you could invest the money at 8%, how much would you have to invest?

Hint: compare your numerical answer with the previous question, would you have to invest more or less than at 6%?

BlackTom AI · Accepted Answer

Present value formula PV = FV / (1 + r)^n applies, because the time value of money tells us a future sum is worth less today by the discount factor (1 + r)^n.

Relevant concepts:
- Time value of money: a dollar today is worth more than a dollar in the future.
- Present value (PV) of a single future amount FV after n periods at interest rate r: PV = FV / (1 + r)^n.

Step-by-step calculation (all intermediate steps shown):
1. Identify the future value and parameters. Here FV = $15,000, r = 8% = 0.08, n = 3 periods.
2. Compute the accumulation factor (1 + r)^n:
   - 1 + r = 1 + 0.08 = 1.08
   - (1.08)^2 = 1.08 × 1.08 = 1.1664
   - (1.08)^3 = 1.1664 × 1.08 = 1.259712
3. Compute PV = FV / (1 + r)^n:
   - PV = 15,000 / 1.259712
   - PV ≈ 11,907.4802
4. Round to cents: PV ≈ $11,907.48.

Why this is correct:
- The formula discounts the future $15,000 back three years at an 8% annual rate. Dividing by (1.08)^3 exactly reverses the compounding that would grow a present amount into $15,000 in three years.
- Because 8% is larger than 6%, the required present investment is smaller than it would be at 6% (higher return means you need to invest less now to reach the same future sum). For comparison, at 6% the PV would be 15,000/(1.06)^3 ≈ $12,590.00, which is larger, as expected.

Why other answers would be incorrect:
- Any answer not equal to 15,000/(1.08)^3 is incorrect because it does not correctly apply the present value formula with the given r and n.
- Using 6% or the wrong exponent (e.g., n ≠ 3) produces different, incorrect amounts.

Conclusion: The correct present amount to invest at 8% is $11,907.48, which matches the provided answer.

If you could invest the money at 8%, how much would you have to invest? Hint: compare your numerical answer with the previous question, would you have to invest more or less than at 6%?数值题

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