An account has a rate of $3.8\%$ . Find the effective annual yield if the interest is compounded semiannually. $\newline$ The effective annual yield is $\square\%$ . $\newline$ (Round to the nearest hundredth as needed.)

Full solution

Q. An account has a rate of

3.8\%

. Find the effective annual yield if the interest is compounded semiannually.

\newline

The effective annual yield is

\square\%

\newline

(Round to the nearest hundredth as needed.)

Identify Given Data: Identify the given annual nominal interest rate and the number of compounding periods per year. $\newline$ We have: $\newline$ Nominal annual interest rate $r$ : $3.8\%$ or $0.038$ (as a decimal) $\newline$ Number of compounding periods per year $n$ : $2$ (since the interest is compounded semiannually) $\newline$ We need to calculate the effective annual yield (EAY).
Use EAY Formula: Use the formula for the effective annual yield when interest is compounded more than once per year. $\newline$ The formula for EAY is: $\newline$ EAY = $(1 + \frac{r}{n})^{n} - 1$ $\newline$ Where $r$ is the nominal annual interest rate and $n$ is the number of compounding periods per year.
Calculate EAY: Plug the values into the formula to calculate the effective annual yield. $\newline$ $EAY = (1 + \frac{0.038}{2})^{2} - 1$ $\newline$ $= (1 + 0.019)^{2} - 1$ $\newline$ $= (1.019)^{2} - 1$ $\newline$ $= 1.019 \times 1.019 - 1$ $\newline$ $= 1.038361 - 1$ $\newline$ $= 0.038361$
Convert to Percentage: Convert the effective annual yield from decimal form to percentage and round to the nearest hundredth as needed. $\newline$ $EAY = 0.038361 \times 100\%$ $\newline$ $= 3.8361\%$ $\newline$ Rounded to the nearest hundredth, the EAY is approximately $3.84\%$ .