Question

Joanne deposits $4,300 into a one-year CD at a rate of 2.3%, compounded daily.

Answer 1

now, let's make the assumption that a year has 365 days, so compounding daily for a year is with a compounding period of 365, so

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$4300\\ r=rate\to 2.3\%\to (2.3)/(100)\dotfill &0.023\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{daily, thus 365} \end{array}\dotfill &365\\ t=years\dotfill &1 \end{cases}`

`A=4300\left(1+(0.023)/(365)\right)^(365\cdot 1)\implies A=4300\left( (365023)/(365000) \right)^(365)\implies \underset{ending~balance}{A\approx 4400.04} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \underset{\textit{earned interest amount}}{\approx 4400.04~~ - ~~4300 ~~ \approx ~~ 100.04}~\hfill`

now, let's take a peek at the APY.

`~~~~~~ \textit{Annual Percent Yield Formula} \\\\ ~~~~~~~~~~~~ \left(1+(r)/(n)\right)^(n)-1~\hfill \begin{cases} r=rate\to 2.3\%\to (2.3)/(100)\dotfill &0.023\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{daily, thus 365} \end{array}\dotfill &365 \end{cases} \\\\\\ \left(1+\cfrac{0.023}{365}\right)^(365)-1\implies \left( \cfrac{365023}{365000} \right)^(365)-1 ~~ \approx ~~ 0.0233 ~~ \approx ~~ \stackrel{0.0233* 100}{2.33~\%}`