For each given interest rate, determine which option is better. Your first option is to receive $70,000 today, which will be put in a bank and left to compound for the next 40 years.
The Correct Answer and Explanation is:
To determine whether receiving \$70,000 today is better than an unspecified second option (perhaps receiving a future amount), we need to analyze how much that \$70,000 would grow if invested at different interest rates over 40 years using compound interest.
We’ll use the compound interest formula:
$$
A = P(1 + r)^t
$$
Where:
- $A$ = Future value of the investment
- $P$ = Principal amount (\$70,000)
- $r$ = Annual interest rate (as a decimal)
- $t$ = Time (40 years)
Let’s compute the future value of \$70,000 under different interest rates:
🔹 At 4% Interest:
$$
A = 70,000(1 + 0.04)^{40} = 70,000(4.801) ≈ 336,070
$$
🔹 At 6% Interest:
$$
A = 70,000(1 + 0.06)^{40} = 70,000(10.286) ≈ 720,020
$$
🔹 At 8% Interest:
$$
A = 70,000(1 + 0.08)^{40} = 70,000(21.724) ≈ 1,520,680
$$
Now, to decide which option is better, we compare these results to the future value of the alternative option (which we assume is a fixed future amount). Suppose your second option is to receive \$500,000 in 40 years.
Compare:
- At 4%, \$70,000 today grows to \$336,070 → \$500,000 in 40 years is better
- At 6%, \$70,000 today grows to \$720,020 → \$70,000 today is better
- At 8%, \$70,000 today grows to \$1,520,680 → \$70,000 today is much better
✅ Conclusion:
- If the interest rate is below ~5.6%, it’s better to wait and receive the future amount (e.g., \$500,000).
- If the interest rate is above ~5.6%, taking the \$70,000 today and investing it is better.
- The higher the interest rate, the more valuable money is today due to compounding.
Therefore, the better option depends entirely on the interest rate and the future value of the second option.