Financial Risk Manager Part 1

Explanation

Correct Answer: B - The simple return is always less than the continuously compounded return.

Mathematical Relationship:

• Simple return: $R_{simple} = \frac{P_t - P_{t-1}}{P_{t-1}}$
• Continuously compounded return (log return): $r_{cc} = \ln(1 + R_{simple})$

Key Insight: For any positive simple return $R_{simple} > 0$, the continuously compounded return $r_{cc}$ is always less than $R_{simple}$ because: $r_{cc} = \ln(1 + R_{simple}) < R_{simple}$ This follows from the inequality $\ln(1+x) < x$ for $x > 0$.

Analysis of Other Options:

A. Incorrect - This is the opposite of the true relationship. The continuously compounded return is always less than the simple return, not greater.

C. Incorrect - The approximation error actually increases as the simple return increases. For small returns, $\ln(1+R) \approx R$, but as R grows larger, the difference becomes more significant.

D. Incorrect - Simple returns have a lower bound of -100% (total loss), while continuously compounded returns can go below -100% (approaching negative infinity as the simple return approaches -100%).