Effective Annual Rate (EAR) Problem

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FuturePFP

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Hi everyone,

I'm preparing for my finance final exam. This course has been frustrating as there was no solution manual for the textbook. Here is one of the many questions I cannot get my head around.

John Smith wants to borrow $600 for 20 days from Loan Sharks Inc. They will charge a flat rate of $15 per loan, plus 8% on the principal, plus Effective Annual Rate of 40% on the first $300 borrowed and Effective Annual Rate of 49% on the next $500.

Calculate the cost of the loan for John, and the EAR of the loan.

Cost of The loan:

flat rate: $15
principal rate: 600 x .08 = $48

Ok this is where I'm losing my head. the EAR!!

I'm guessing I need to convert the EAR to a daily rate.

so for the first $300, r = [(1.40) ^ (1/365)] - 1] x 365 = 33.66%

I do the same for the second EAR of 49% but now what.

Please help. Thanks in advance.
 
Last edited:
hn Smith wants to borrow $600 for 20 days from Loan Sharks Inc. They will charge a flat rate of $15 per loan, plus 8% on the principal, plus Effective Annual Rate of 40% on the first $300 borrowed and Effective Annual Rate of 49% on the next $500.

$600 for 20 days

flat rate of $15 per loan ==> $15.00
plus 8% on the principal ==> 600*0.08*(20/365) = $2.63
plus Effective Annual Rate of 40% on the first $300 borrowed ==> 300*(1.4)^(20/365) - 300 = $5.58
Effective Annual Rate of 49% on the next $500. (600-300)*(1.49)^(20/365) - (600-300) = $6.63

$15.00 + $2.63 + $5.58 + $6.63 = $29.84 -- There's your interest for the 20 days.

Now what?
 
Thank You

Thank You very much tkhunny.

If you could just clarify something for me please. When calculating the EAR, why did you subtract the borrowed amount?

"plus Effective Annual Rate of 40% on the first $300 borrowed ==> 300*(1.4)^(20/365) - 300 = $5.58"

"Effective Annual Rate of 49% on the next $500. (600-300)*(1.49)^(20/365) - (600-300) = $6.63"

Really appreciate the help.
 
EAR of the loan

So the final part of the question, the EAR of the loan would be as follows:

= 500 + (interest amount) divided by original loan amount

= 523.53 / 500 = 1.047 or rather 104.71 %
 
I just wanted the Interest. That's a funny thing about Simple Interest vs. Compound Interest.

Simple: I = Prt -- We calculate the interest directly and the new balance is a little harder: N = P+I = New Balance

Compound: N = P(1+i)^t -- We calculate the new balance directly and the interest is a little harder: I = N - P = Interest
 
Hmmm...the 'A' is for Annual. Okay I obviously calculated the wrong EAR. Confused...
 
Well, it would help to use $600, rather than $500... :D

There are so many ways that we see "interest" in this problem, it is a bit of a challenge to decide how to change that to an Annual Rate.

Simple: 20 days: 29.84 = 600.00*r*(20/365) ==> r = 0.9076333 = 90.76%

Compound: 20 days: \(\displaystyle 629.84 = 600.00*(1+i)^{20/365}\) ==> i = 1.424887 = 142.49%

Suffice it to say that the annual rate is a VERY LARGE number. This is why we have Consumer Protection laws.
 
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