Question

Melanie is looking for a loan. She is willing to pay no more than an effective rate of 9. 955% annually. Which, if any, of the following loans meet Melanie’s criteria? Loan A: 9. 265% nominal rate, compounded weekly Loan B: 9. 442% nominal rate, compounded monthly Loan C: 9. 719% nominal rate, compounded quarterly a. B only b. A and C c. A and B d. None of these fit Melanie’s criteria. Please select the best answer from the choices provided A B C D.

Answer 1

Melanie should take A and B, being r(loan a)= 9.699% annually, r(loan b)=9.862% annual

What options Melanie should choose for the best deal?

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.

$r(Eweek)=\dfrac{0.09265}{52} =0.00178173$

Or 0.178173% effective weekly. Now we can transform it into an effective annual rate.

$r(e,a)=1+(r(e.week))^{52} -1$

$r(e,a)=1+(0.00178173)^{52} -1=0.09699$

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.

$r(Emonth)=\dfrac{0.09442}{12} =0.00786833$

Or 0.786833% effective monthly, let´s turn it into an effective annual rate.

$r(e,a)=1+(r(e.month))^{12} -1$

$r(e,a)=1+(0.00786833))^{12} -1=0.09862$

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.

$r(Eweek)=\dfrac{0.09719}{4} =0.0242975$

or 2.42975% effective quarterly, now, let´s convert it into an effective annual rate.

$r(e,a)=1+(0.0 242975))^{12} -1=0.010079$

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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