if the nominal annual interest rate is 24% compounded continuously, then the effective interest rate per…

if the nominal annual interest rate is 24% compounded continuously, then the effective interest rate per month is\n\na. $\frac{1}{12}(e^{0.24}-1)$\nb. $e^{0.02}-1$\nc. $e^{0.24}-1$\nd. $12\times(e^{0.02}-1)$\ne. $e^{0.24/30}-1$

if the nominal annual interest rate is 24% compounded continuously, then the effective interest rate per month is\n\na. $\frac{1}{12}(e^{0.24}-1)$\nb. $e^{0.02}-1$\nc. $e^{0.24}-1$\nd. $12\times(e^{0.02}-1)$\ne. $e^{0.24/30}-1$

Answer

Explanation:

Step1: Recall continuous - compounding formula

The formula for the effective - annual rate (EAR) when the nominal annual rate $r$ is compounded continuously is $EAR = e^{r}-1$. Given $r = 0.24$, the effective - annual rate is $e^{0.24}-1$.

Step2: Find the effective monthly rate

Let the effective monthly rate be $i$. If we compound the monthly rate $i$ 12 times in a year, we should get the effective - annual rate. Let $i$ be the effective monthly rate. Then, $(1 + i)^{12}=e^{0.24}-1$. Also, if we consider the continuous - compounding approach for monthly rate, we know that the nominal annual rate $r = 0.24$, so the nominal monthly rate $r_m=\frac{0.24}{12}=0.02$. The formula for the effective monthly rate when compounded continuously is $i = e^{r_m}-1$. Substituting $r_m = 0.02$ into the formula, we get $i=e^{0.02}-1$.

Answer:

B. $e^{0.02}-1$