Question

Robert just got a new credit card and immediately made a purchase for $2750. The card offers a 0% APR for the first 60 days and a 17.99% APR afterward, compounded daily. Robert doesn't expect to pay off the $2750 balance on the card for one year, nor does he expect to make any more purchases during the year. He wants to know how much money in interest the 0% APR for the first 60 days will save him. Help Robert calculate the answer. Ignore any possible late payment fee For how many days out of the year will Robert pay interest at a 17.99% APR? What is the effective interest rate offered by Robert's credit card? Round your answer to two decimal places. How much will Robert pay in interest on the $2750 purchase over the course of the year? What would the effective interest rate have been if the APR had been 17.99%, compounded daily, for the whole year? Round your answer to two decimal places. How much would Robert have paid in interest on the $2750 purchase over the course of the year with the effective interest rate from Part IV? How much money in interest will the 0% APR for the first 60 days save Robert?

Answer 1

Given that Robert just got a new credit card and immediately made a purchase for $2,750. The card offers a 0% APR for the first 60 days and a 17.99% APR afterward, compounded daily. Robert doesn't expect to pay off the $2,750 balance on the card for one year, nor does he expect to make any more purchases during the year.

PART 1:
Given that Robert's new credit card offers a 0% APR for the first 60 days, thus the number of days out of the year that Robert will pay interest at a 17.99% APR is given by 365 - 60 = 305 days.


PART 2:
The effective interest rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.
The effective interest rate is calculated by dividing the norminal rate (expressed as decimal) by the number of compounding in a year, add the result to one, raise the result to the power of the number of compounding in a year and then subtract one from the result.
i.e. The effective interest rate is given by
$r=\left(1+ \frac{i}{n} \right)^n-1$
where: i is the given APR; n is the number of compounding in the year.

Therefore, the effective interest rate offered by Robert's credit card is given by
$r=\left(1+ \frac{0.1799}{305} \right)^{305( \frac{305}{365} )}-1 \\ \\ =(1+0.000589836)^{ \frac{18,605}{73} }-1 \\ \\ =(1.000589836)^{\frac{18,605}{73}}-1 \\ \\ =1.162163-1=0.162163 \\ \\ =16.22\%$
[we multiplied the exponent by $\frac{305}{365}$ because the interest was paid for only 305 days out of the 365 days of the year.


PART 3:
The amount Robert pays in interest on the $2,750 purchase over the course of the year is given by
$\ 2,750\times0.162163=\ 445.95$


PART 4:
The effective interest rate is given by
$r=\left(1+ \frac{i}{n} \right)^n-1$
where: i is the given APR; n is the number of compounding in the year.


If the APR had been 17.99%, compounded daily, for the whole year, n = 365

Therefore, the effective interest rate had it been that the APR is 17.99%, compounded daily, for the whole year is given by
$r=\left(1+ \frac{0.1799}{365} \right)^{365}-1 \\ \\ =(1+0.000492877)^{365}-1 \\ \\ =(1.000492877)^{365}-1 \\ \\ =1.197045-1=0.197045 \\ \\ =19.70\%$


PART 5:
The amount Robert would have paid in interest on the $2,750 purchase over the course of the year with the effective interest rate from Part IV is given by
$\ 2,750\times0.197045=\ 541.87$

PART 6:
The amount of money in interest that the the 0% APR for the first 60 days will save Robert is given by $541.87 - $445.95 = $95.92