when calculating the effective rate of a loan, which statement or statements must be true if n is greater…

Explanation:

Step1: Recall effective - rate formula

The formula for the effective - annual rate (EAR) 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.

Step2: Analyze statement I

The variable $n$ represents the number of compounding periods per year, not the length of the loan. For example, if $n = 2$, it means semi - annual compounding, and the loan could be for less than a year. So, statement I is false.

Step3: Analyze statement II

Using the formula $EAR=(1+\frac{r}{n})^{n}-1$, when $n>1$, by the binomial expansion $(1 + x)^{n}=1+nx+\frac{n(n - 1)}{2!}x^{2}+\cdots+x^{n}$ ($x=\frac{r}{n}$), we have $(1+\frac{r}{n})^{n}=1 + r+\frac{n(n - 1)}{2}\frac{r^{2}}{n^{2}}+\cdots>1 + r$. Since the nominal rate is $r$ and $EAR=(1+\frac{r}{n})^{n}-1$, the effective rate will exceed the nominal rate when $n>1$. So, statement II is true.

Step4: Analyze statement III

$n>1$ does not necessarily mean monthly compounding. $n$ could be 2 (semi - annual), 4 (quarterly), etc. So, statement III is false.

Answer:

a. II only