Question

If you have a choice of borrowing money at 8% compounded semi-annually or borrowing at 8.15% compounded annually, which should you choose and why? Please go into detail

Answer 1

one simple way to take a peek at this is by looking at the APY value, namely the Annual Percent Yield for each of the choices, and use the one that gives you the higher APY, higher APY, higher bucks.

`~~~~~~ \textit{Annual Percent Yield Formula} \\\\ ~~~~~~~~~~~~ \left(1+(r)/(n)\right)^(n)-1 \\\\ \textit{for the 8\%} \begin{cases} r=rate\to 8\%\to (8)/(100)\dotfill &0.08\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semi-annually, thus twice} \end{array}\dotfill &2 \end{cases} \\\\\\ \left(1+(0.08)/(2)\right)^(2)-1\implies (1.04)^2-1\implies 0.0816~\hfill 8.16\%`

`\textit{for the 8.15\%} \begin{cases} r=rate\to 8.15\%\to (8.15)/(100)\dotfill &0.0815\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1 \end{cases} \\\\\\ \left(1+(0.0815)/(1)\right)^(1)-1\implies (1.0815)-1\implies 0.0815~\hfill 8.15\%`

well, clearly the first one has a higher APY, so you'd want that one.