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.