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Melanie should take A and B, being r(loan a)= 9.699% annually, r(loan b)=9.862% annual
It is given that
Loan A: 9.265% nominal rate, compounded weekly
In order to find the easiest effective rate, we need to divide this rate by 52 (which are the weeks in a year). Once we do that, we convert this effective weekly rate into an effective annual rate. Let´s walk you through all this.
[tex]r(Eweek)=\dfrac{0.09265}{52} =0.00178173[/tex]
Or 0.178173% effective weekly. Now we can transform it into an effective annual rate.
[tex]r(e,a)=1+(r(e.week))^{52} -1[/tex]
[tex]r(e,a)=1+(0.00178173)^{52} -1=0.09699[/tex]
Or 9.669% annual, which is less than 9.955%, so this one is selected, let´s check the next.
Loan B: 9.442% nominal rate, compounded monthly
Just like we did with Loan A, we need to divide this rate too, only this time, we will divide by 12, therefore obtaining an effective monthly rate.
[tex]r(Emonth)=\dfrac{0.09442}{12} =0.00786833[/tex]
Or 0.786833% effective monthly, let´s turn it into an effective annual rate.
[tex]r(e,a)=1+(r(e.month))^{12} -1[/tex]
[tex]r(e,a)=1+(0.00786833))^{12} -1=0.09862[/tex]
Or 9.862% annual, so this rate would work for Melanie too. This means that option C) is the answer we are looking for but, let´s walk that extra mile and turn that Loan C rate into an annual rate.
[tex]r(Eweek)=\dfrac{0.09719}{4} =0.0242975[/tex]
or 2.42975% effective quarterly, now, let´s convert it into an effective annual rate.
[tex]r(e,a)=1+(0.0 242975))^{12} -1=0.010079[/tex]
That is 10.079% effective annual, therefore, Loan C is not an option for Melanie.
Thus Melanie should take A and B, being r(loan a)= 9.699% annually, r(loan b)=9.862% annual
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