Question

Correct me !
21.34% divided by 365
=0.000584658 ?
Sydney possesses a credit card with an APR of 21.34% compounded daily. What is the APY on this credit card?

5.85% <------

23.78%

28.84%

47.56%

Answer 1

21.34% = 21.34% * 1/100% = 0.2134
0.2134/365 = 0.000584658

however, you have fallen into a trap. frequently the answers on multiple choice questions will include wrong answers that you might get if you don't understand what you are doing. This is one of them.

Think about it: with annual compounding, the interest would be 21.34% of the balance. With more frequent compounding, the total interest gets a bit larger (how much larger depending on how often the compounding happens). With daily compounding, each day, 0.000584658 * the balance gets tacked on as interest. Let's say we had a balance of $1000. The first day, interest = $1000*0.000584658 = 58.4658 cents. The next day, it will be $1000.584658 * 0.000584658 = 58.5 cents. Over the course of a year, we're going to have at least 365 * 58.5 cents in interest, which adds up to $213 and change. Remember, that was interest on a starting balance of $1000, so that means at a minimum our interest rate must be 21.3%. 5.85% is just out of the question.

The formula for computing compound interest can be written as
\[PV(1+\frac{r}{n})^{nt}\]where \(r\) is the annual rate, expressed as a decimal; \(n\) is the number of compounding periods per year; \(t\) is the number of years, and \(PV\) is the balance or present value. If we plug in our numbers, we get
\[PV(1+\frac{0.2134}{365})^{1*365}=1.2378 *PV\]What does that suggest the answer should be?