Explanation

To convert a simple return to a continuously compounded return, we use the relationship:

$`$1` + R_t = e^{r_t}$$

Where:

• $R_t$ = simple return (15% or 0.15)
• $r_t$ = continuously compounded return

Solving for $r_t$:

$r_t = \ln(1 + R_t) = \ln(1.15)$

Calculating this:

$r_t = \ln(1.15) = 0.1398 = 13.98\%$

Therefore, the equivalent continuously compounded return is 0.1398 or 13.98%.

This conversion is important in quantitative finance because continuously compounded returns have several advantages:

• They are additive over time
• They are normally distributed in many financial models
• They are used in options pricing models like Black-Scholes