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If I lend you $100 at, say, 10% annual interest for a 3 year term (36 months), your monthly instalment (usual amortization schedule) should be $3 per month for 36 months, for a total of $116, right?

My question is how to back out an effective rate of return if you don't repay in full. E.g. say you only pay $1/month (giving a total repayment of $36) - what is my rate of return (as lender)?

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  • Depends on the loan. Loans like that may be "pay interest, pay back capital in 3 years".
    – TomTom
    Commented Jul 20, 2018 at 10:12

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With

r is the monthly rate
n is the number of months
s is the principal
d is the monthly payment

Assuming the annual rate is nominal 10% compounded monthly

r = 10/100/12
n = 36
s = 100

Using the ordinary annuity formula

d = (r (1 + r)^n s)/((1 + r)^n - 1) = 3.22672

resulting in a repayment total of 36 * d = $116.16

Resetting the monthly repayment

d = 1

Numerically solving the ordinary annuity formula for r

r = -0.0473652

The annual nominal rate is 12 * r = -56.8383 % compounded monthly

Annuity formula solution

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For more information on annuities see Present & Future Value of Annuities

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