Question

What rate of interest compounded quarterly will yield an effective interest rate of 7%?

Short answer

The rate of interest that when compounded quarterly will yield an effective interest rate of 7% is 6.823%

Step-by-step solution

Step 1. Given Information

Given to determine the rate of interest that when compounded quarterly will yield an effective interest rate of 7%

Step 2. Calculation

The effective rate of interest r_e for an investment with rate of interest r and compounding n times per year is given by $r_e={\left(1+\frac{r}{n}\right)}^n-1$

Here the investment is compounded quarterly. So $n=4$

And since the effective rate of interest is 7%, the decimal form is given by role="math" localid="1646650286759" $r_e=0.07$

Plugging the values:

role="math" localid="1646650531814" $$\begin{gathered} r_e={\left(1+\frac{r}{n}\right)}^n-1 \\ 0.07={\left(1+\frac{r}{4}\right)}^4-1 \\ {\left(1+\frac{r}{4}\right)}^4=0.07+1 \\ {\left(1+\frac{r}{4}\right)}^4=1.07 \\ 1+\frac{r}{4}=\sqrt[4]{1.07} \\ 1+\frac{r}{4}\cong 1.017058 \\ \frac{r}{4}=1.017058-1 \\ \frac{r}{4}=0.017058 \\ r=0.06823 \end{gathered}$$

Step 3. Conclusion

The rate of interest that when compounded quarterly will yield an effective interest rate of 7% is 6.823%