Bob has an account with $1000 (U.S.). He converts to $F and makes 15% continuous in one year in the Freedonia Stock Market. After this one year the currency price is now 4 that is $F 4 per $1 U.S. He converts back to U.S dollars at the rate of exchange. What amount (in U.S dollars) did he make or lose in the one year? What was his continuous rate of return?

Let's break down the problem and solve it step by step.

### 1. Initial Conversion to Freedonia Currency ($F$)
Bob starts with $1000 USD. The exchange rate when he converts to Freedonia's currency $F$ is $F4 per $1 USD.

\[
\text{Amount in } F = 1000 \times 4 = 4000 F
\]

### 2. Growth in Freedonia Stock Market
Bob earns a continuous rate of return of 15% over one year. The formula for continuous compounding is:

\[
A = P \times e^{rt}
\]

Where:
- $P$ is the initial amount in $F$ (which is 4000 $F$),
- $r$ is the continuous rate of return (0.15),
- $t$ is the time in years (1 year),
- $e$ is the base of the natural logarithm.

Substituting the values:

\[
A = 4000 \times e^{0.15 \times 1} = 4000 \times e^{0.15} \approx 4000 \times 1.1618 \approx 4647.2 F
\]

### 3. Conversion Back to U.S. Dollars
After one year, the exchange rate is still $F4 per $1 USD. Now, Bob wants to convert back to USD.

\[
\text{Amount in USD} = \frac{4647.2 F}{4} \approx 1161.8 \text{ USD}
\]

### 4. Calculate the Gain or Loss
Bob started with $1000 USD. After conversion back, he has approximately $1161.8 USD.

\[
\text{Net Gain or Loss} = 1161.8 - 1000 = 161.8 \text{ USD}
\]

So, Bob made a gain of approximately **$161.8** USD in one year.

### 5. Continuous Rate of Return in USD
To find the continuous rate of return in USD, we use the formula again, but this time based on the initial and final amounts in USD.

Given:
- $P_{\text{USD}} = 1000$ USD (initial amount),
- $A_{\text{USD}} = 1161.8$ USD (final amount after one year),
- $t = 1$ year.

The formula for the continuous rate of return $r_{\text{USD}}$ is:

\[
A_{\text{USD}} = P_{\text{USD}} \times e^{r_{\text{USD}} \times t}
\]

\[
1161.8 = 1000 \times e^{r_{\text{USD}}}
\]

\[
e^{r_{\text{USD}}} = \frac{1161.8}{1000} = 1.1618
\]

Taking the natural logarithm of both sides:

\[
r_{\text{USD}} = \ln(1.1618) \approx 0.15 \text{ or } 15\%
\]

Thus, Bob's continuous rate of return in USD is **15%**.

### Summary:
- **Amount gained**: $161.8 USD
- **Continuous rate of return in USD**: 15%

Would you like more details or have any questions? 

Here are some related questions:
1. How would the result change if the exchange rate had fluctuated during the year?
2. What would be the result if Bob had a different continuous rate of return, say 10%?
3. How do we calculate the discrete compounding rate of return instead of the continuous rate?
4. What would happen if Bob invested in another country's stock market with a different exchange rate?
5. How does continuous compounding differ from simple interest?

**Tip:** When dealing with international investments, it's essential to consider both the performance of the investment and the currency exchange rates, as both can significantly impact the final outcome.

Learn how to calculate the return on an investment in a foreign stock market using currency conversion and continuous compounding. This problem involves converting $1000 USD to Freedonia currency, applying a 15% continuous growth rate over one year, and converting back to USD. The final gain is $161.8 with a 15% continuous rate of return.

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