select the correct answer. what is the annual interest rate for an account that compounds monthly at a rate…

select the correct answer. what is the annual interest rate for an account that compounds monthly at a rate of 2.5%?\na. 2.10%\nb. 2.50%\nc. 2.53%\nd. 2.55%\ne. 2.58%

select the correct answer. what is the annual interest rate for an account that compounds monthly at a rate of 2.5%?\na. 2.10%\nb. 2.50%\nc. 2.53%\nd. 2.55%\ne. 2.58%

Answer

Explanation:

Step1: Recall compound - interest formula for effective annual rate

The formula for the effective annual rate (EAR) when compounding $n$ times a year is $EAR=(1 + \frac{r}{n})^{n}-1$, where $r$ is the nominal annual interest rate and $n$ is the number of compounding periods per year. Here, the monthly interest rate $i = 2.5%=0.025$, and $n = 12$. We want to find the nominal annual rate $R$. Since the monthly rate $i=\frac{R}{n}$, then $R = n\times i$.

Step2: Calculate the annual interest rate

Since the monthly rate is $2.5%$ and there are 12 months in a year, the annual interest rate $R=12\times2.5% = 30%$. But it seems there is a mis - understanding in the problem setup. If the monthly rate is $r_m=0.025$ (2.5% per month), the effective annual rate $EAR=(1 + 0.025)^{12}-1$. [ \begin{align*} EAR&=(1 + 0.025)^{12}-1\ &=1.025^{12}-1\ &\approx1.344888824 - 1\ &=0.344888824\approx34.49% \end{align*} ] If the problem means that the nominal annual rate $r$ is divided by 12 to get a monthly rate of 2.5%, then the nominal annual rate $r = 12\times2.5%=30%$. But if we assume the problem is asking for the effective annual rate when the nominal rate is divided such that the monthly rate is 2.5%: [EAR=(1+\frac{r}{n})^{n}-1=(1 + 0.025)^{12}-1\approx 0.3449] If the problem is just asking for the nominal annual rate given a 2.5% monthly rate, the nominal annual rate $R = 12\times2.5%=30%$ which is not in the options. If we assume the problem is mis - stated and the 2.5% is the nominal annual rate divided by 12: The nominal annual rate $r=12\times2.5% = 30%$. If we assume the problem is asking for the effective annual rate when the nominal rate $r$ gives a monthly rate of 2.5%: [EAR=(1 + 0.025)^{12}-1\approx0.3449] If we assume the problem means the nominal annual rate: Since the monthly rate is 2.5%, the nominal annual rate $r = 12\times2.5%=30%$ (not in options). If we assume the problem has a wrong setup and we consider the following: Let's assume the 2.5% is the nominal annual rate. The effective annual rate $EAR=(1+\frac{0.025}{12})^{12}-1$ [ \begin{align*} EAR&=(1+\frac{0.025}{12})^{12}-1\ &\approx(1 + 0.00208333)^{12}-1\ &\approx1.00208333^{12}-1\ &\approx1.02528 - 1\ &=0.02528\approx2.53% \end{align*} ]

Answer:

C. 2.53%