what is the effective interest rate of 5% interest compounded monthly (round to the nearest.01%)?\n5.12\nques…

Question 1

Explanation:

Step1: Recall effective - rate formula

The formula for the effective - annual rate (EAR) when compounded $n$ times a year is $EAR=(1 + \frac{r}{n})^{n}-1$, where $r$ is the annual interest rate and $n$ is the number of compounding periods per year. Here, $r = 0.05$ and $n=12$ (monthly compounding). $EAR=(1+\frac{0.05}{12})^{12}-1$

Step2: Calculate the value

First, calculate $1+\frac{0.05}{12}=1+\frac{1}{240}=\frac{240 + 1}{240}=\frac{241}{240}\approx1.004167$. Then, $(1.004167)^{12}\approx1.051162$. So, $EAR = 1.051162-1=0.051162\approx5.12%$

Answer:

$5.12%$

Question 3

Explanation:

Step1: Recall continuous - compounding effective - rate formula

The formula for the effective - annual rate when compounded continuously is $EAR = e^{r}-1$, where $r$ is the annual interest rate. Here, $r = 0.052$. $EAR=e^{0.052}-1$

Step2: Calculate the value

We know that $e^{0.052}\approx1.05338$. So, $EAR=1.05338 - 1=0.05338\approx5.34%$

Answer:

$5.34%$

Question 4

Explanation:

Step1: Recall present - value formula for compound interest

The present - value formula is $PV=\frac{FV}{(1+\frac{r}{n})^{nt}}$, where $FV$ is the future value, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is the number of years. Here, $FV = 50000$, $r=0.04$, $n = 2$ (semi - annual compounding), and $t = 3$. $PV=\frac{50000}{(1+\frac{0.04}{2})^{2\times3}}$

Step2: Simplify the denominator

First, calculate $1+\frac{0.04}{2}=1 + 0.02=1.02$. Then, $(1.02)^{6}=1.02\times1.02\times1.02\times1.02\times1.02\times1.02\approx1.126162$.

Step3: Calculate the present value

$PV=\frac{50000}{1.126162}\approx44392.53$

Answer:

$$44392.53$