Assume that in 2020, a Liberty Seated half dollar issued in 1890 was sold for $215,000. What was the rate of return on this investment?

To find the rate of return on the investment, we use the formula for compound interest:

\[
A = P \left(1 + r\right)^t
\]

Where:
- $ A $ is the amount of money after $ t $ years (which is $215,000).
- $ P $ is the original amount invested (the value of the half dollar in 1890).
- $ r $ is the annual rate of return.
- $ t $ is the number of years the money was invested (from 1890 to 2020, which is 130 years).

We need to solve for $ r $. Rearranging the formula:

\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]

Since we do not know the original value $ P $, the problem suggests that we find $ r $ using the formula:

\[
r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]

We assume $ P $ to be some value in 1890, but the key interest here is the rate $ r $. The formula simplifies into the form of the rate of return, assuming that the coin was purchased at a very low value initially (or nominal value). Let’s compute the rate of return assuming $ P $ was nominal.The rate of return on the investment is approximately **10.49%** per year over the 130-year period.

Would you like to explore this calculation further or have any other questions?

Here are some related questions you might consider:
1. What if the coin was purchased for $1 instead of 50 cents? How would the rate change?
2. How does this rate of return compare with other types of investments over a similar period?
3. What would the coin be worth in 2050 if this rate of return continued?
4. How much would you need to invest today at this rate to reach $1,000,000 in 50 years?
5. What if the coin had been sold in 1950? What would the rate of return have been then?

**Tip:** When considering historical investments, the rate of return can provide valuable insight into the long-term performance and potential growth of similar investments over time.

Learn how to calculate the rate of return on an investment using compound interest. Explore the case study of a Liberty Seated half dollar purchased in 1890 and sold for $215,000 in 2020, yielding a rate of return of approximately 10.49% per year over 130 years.

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