If the risk-free rate is 6.5 percent compounded annually, what is the continuously compounded risk-free rate equal to?
The correct answer and explanation is:
To convert an annually compounded interest rate to a continuously compounded rate, we can use the following relationship between the two: rcc=ln(1+rac)r_{cc} = \ln(1 + r_{ac})
Where:
- rccr_{cc} is the continuously compounded rate.
- racr_{ac} is the annually compounded rate.
- ln\ln represents the natural logarithm.
Given that the annually compounded risk-free rate is 6.5%, or 0.065 in decimal form, the calculation becomes: rcc=ln(1+0.065)r_{cc} = \ln(1 + 0.065) rcc=ln(1.065)r_{cc} = \ln(1.065) rcc≈0.0629 or 6.29%r_{cc} \approx 0.0629 \text{ or } 6.29\%
Thus, the continuously compounded risk-free rate is approximately 6.29%.
Explanation:
The reason for using this conversion formula is that continuously compounded rates assume an infinitely frequent compounding period. In contrast, annually compounded rates assume a single compounding period per year. The formula effectively translates the annual compounding effect into the continuous compounding format.
The natural logarithm (ln\ln) is used because continuously compounded interest is based on exponential growth. The formula shows how much an annually compounded rate would grow in the same time frame when compounded continuously. By taking the natural logarithm of 1+rac1 + r_{ac}, the formula converts the discrete compounding process into a continuous one, yielding a lower rate compared to the nominal annual rate.