How can I calculate minimum payment on quick loan? [closed]

Question

I am working on calculator for quick loans. Client can select how much he will pay monthly. Base on it payment schedule will be calculated. Payment interest depends on 1 payment period. Number of payments is limited. How can I calculate what is a minimum monthly payment if I know that number of payments should be 5, amount of loan is 250, APR is 70% and payment periods are next:

• 1 payment - 9 days,
• 2 payment - 30 days
• 3 payment - 28 days
• 4 payment - 30 days
• 5 payment - 32 days

Is it possible to calculate minimum based on this logic?

Answer 1

APR is 70%, compounded monthly (assumed). First convert to effective annual rate

ear = (1 + 0.7/12)^12 - 1 = 0.9746 = 97.46 %

The term of the loan is 129 days, so the interest rate for the term is

i = (1 + ear)^(129/365) - 1 = 0.2718 = 27.18 %

The log rate is

r = ln(1 + i) = 0.2405

Without any repayments the amount owed at the end of the loan term would be

250 e^r = 317.955

which can also be calculated from

250 (1 + i) = 317.955

For equal payments

-250 e^r + p e^((120/129)r) + p e^((90/129)r) + p e^((62/129)r) + p e^((32/129)r) + p = 0

therefore -317.955 + 5.61728 p = 0

therefore p = 56.6031

giving a total cost of loan of 5 * 56.6031 = 283.015

However, if the payments do not have to be equal, the cost of the loan can be minimised by paying it all off in the first installment

-250 e^r + p e^((120/129)r) = 0

giving a single repayment on day 9 of p = 254.229

Check

The daily rate is

d = (1 + ear)^(1/365) - 1 = 0.0018657 = 0.18657 %

Cost of loan paid off after 9 days is 250 (1 + d)^9 = 254.229