Question

Janet took a $3000 loan from an online lending platform. The platform charges a "processing fee" of $400 and only pays Janet $2600 right now. Janet needs to pay back the $3000 debt in 3 monthly installments （equal amount). What is the implied interest rate (in terms of APR) for this loan?
A. 238%
B. 90%
C. 78%
D. 214%

Answer 1

• Janet has received onlt $2600 right now. But she has to pay $3000 in 3 equal monthly installments. This means that the difference $400 is towards interest payment.
• As she has to repay the loan by paying $3000 in 3 equal monthly installments, the monthly installment comes to $1000 ($3000/3).
• We can calculate the implied interest rate using the Annuity Formula, which is
  • PV = P [ 1 - (1 + r)^-n ] / r , where
    • PV = Present Value of loan taken = $2600 ( as Janet has received only $2600 effectively, as the processing fee of $400 has already got deducted from the loan amount of $3000 before the disbursement)
    • P = Periodic (monthly, in our case) payments = $1000 (which is the amount of equal monthly installment)
    • r = Rate of interest per period = APR / No. of months
    • n = No.of periods = 3 (months)

• Therefore, 2600 = 1000 x [1 - (1+r)^-3 ] / r
  • 2.6 =[ 1 - (1+r)^-3 ] / r
  • In the following table, which ever option gives [ 1 - (1+r)^-3 ] / r = 2.6, that is our answer

Options	r = (APR/12 months) / 100	[ 1 - (1+r)-3 ] / r
A = 238%	r = (238/12)/100 = 0.19833	2.112
B = 90%	r = (90/12)/100 = 0.075	2.600
C = 78%	r = (78/12)/100 = 0.065	2.6484
D = 214%	r = (214/12)/100 = 0.17833	2.1801

Hence, the implied interest rate is 90% (APR terms). Correct answer is Option B.

Alternative method:

If you want to calculate the rate using excel,

use the formula PMT(rate,nper,pv) , and apply all of these rates to the formula. PMT gives the Equated monthly installment amount. So, the rate at which the answer on applying this formula comes to 1000, that is your answer.

Here, rate = r calculated in table above ; nper = number of periods, that is 3 ; pv = present value of the loan = 2600

For eg. =PMT(0.075,3,2600,0) gives you the answer 1000 (approx).

I hope I have helped you with the concept.