Answer: ln(1 + R)

## Explanation The continuously compounded rate of return is derived from the relationship between discrete and continuous compounding. ### Key Formula: For a holding period return R (discrete return), the continuously compounded rate of return r is given by: $$1 + R = e^r$$ Taking the natural logarithm of both sides: $$r = \ln(1 + R)$$ ### Why this is correct: - **Option A (eᴿ − 1)**: This is incorrect because eᴿ − 1 gives the discrete return equivalent to a continuously compounded rate of R, not the other way around. - **Option B (ln(1 + R))**: This is correct as shown above. - **Option C (ln(1 + R) − 1)**: This is incorrect; subtracting 1 would not give the proper continuously compounded rate. ### Example: If an investment has a holding period return of 10% (R = 0.10), then: - Continuously compounded rate = ln(1 + 0.10) = ln(1.10) ≈ 0.0953 or 9.53% This makes sense because continuous compounding at 9.53% gives the same ending value as discrete compounding at 10% over the same period.

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