# Rate of return on a loan

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)?

• Depends on the loan. Loans like that may be "pay interest, pay back capital in 3 years". – TomTom Jul 20 '18 at 10:12

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

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