Tabansi

2021-08-12

Find the effective interest rate ${r}_{e}$ for investments with the following annual interest rates.
Express your answer as a percentage, rounded to 2 decimals, and without the  sign (e.g., 8.79)
a) $3.55\mathrm{%}$ compounded monthly
b) $3.53\mathrm{%}$ compounded daily (using 365 days per year).
c) $3.49\mathrm{%}$ compounded continuously

Tuthornt

Step 1
Here we have to find the effective interest rate for investment with the following interest rate,
So, we will use the following formula,
${r}_{e}={\left(1+\frac{r}{n}\right)}^{n}-1$
Or, ${r}_{e}={e}^{r}-1$
Here it is depending on if the interest is compounded n times per year or continuously.
Step 2
a) $3.55\mathrm{%}$ compounded monthly
So, here we use the formula, ${r}_{e}={\left(1-\frac{r}{n}\right)}^{n}-1$
Where, $r=0.0355$ and $n=12$
So, ${r}_{e}={\left(1+\frac{0.0355}{12}\right)}^{12}-1$
${r}_{e}={\left(1.002958\right)}^{12}-1$
${r}_{e}=0.03608$
Hence, the effective interest rate is about $3.61\mathrm{%}$
Step 3
b) $3.53\mathrm{%}$ compounded daily (using 365 days per year).
Using the formula, ${r}_{e}={\left(1+\frac{r}{n}\right)}^{n}-1$
Where, $r=0.0353$ and $n=365$
So, ${r}_{e}={\left(1+\frac{0.0353}{365}\right)}^{365}-1$
${r}_{e}=0.03592$
Hence, the effective interest rate is about $3.59\mathrm{%}$
Step 4
c) $3.49\mathrm{%}$ compounded continuously
Here, we are using the formula ${r}_{e}={e}^{r}-1$
Where, $r=0.0349$
${r}_{e}={e}^{0.0349}-1$
${r}_{e}=0.03551$
Hence, the effective interest rate is about $3.55\mathrm{%}$

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