Question

A local finance company quotes an interest rate of 18 percent on one-year loans. So, if you borrow $35,000, the interest for the year will be $6,300. Because you must repay a total of $41,300 in one year, the finance company requires you to pay $41,300/12, or $3,441.67, per month over the next 12 months.

a. what rate would legally have to be quoted?

b. what is the effective annual rate?

Answer 1

(a)  
Amount of loan (P)=   35000
  
Total months (n)=   12
Equal monthly Payment=   $3,441.67

Equal payments formula = P* i *((1+i)^n)/(((1+i)^n)-1)  
3441.67 = 35000*i*((1+i)^12)/(((1+i)^12)-1)  

We will Assume ínterest rate per month is 3%  
Equal Payment=
35000*0.03*((1+0.03)^12)/(((1+0.03)^12)-1)  
=3516.172992  
  
We will Assume ínterest rate per month is 2.5%

Equal Payment=
35000*0.025*((1+0.025)^12)/(((1+0.025)^12)-1)  
=3412.049445  

We will calculate (i) by interpolation formula  
interpolation formula = uper rate + (uper rate - lower rate)*(Uper amount - actual Payment)/(uper amount - lower amount)  
=0.03 - ((0.03-0.025)*(3516.172992-3441.67)/(3516.172992-3412.049445)
=0.003- 0.003577624569
=0.02642237543   or 2.642237543%
Per month rate is 2.642237543%  

So annual Rate legally have to be Quoted is 2.642237543%*12=   31.71%

(b)  
Effective annual rate =((1+monthly rate)^(no of months in a year))-1  
((1+   0.02642237543
0.3674559024   or 36.75%
  
So EAR of Loan is 36.75%   
  

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