I am trying to solve this question: Your brother-in-law asks you to lend him $100,000 as a second mortgage on his vacation home. He promises to make level monthly payments for 10 years, 120 payments in all. You decide that a fair interest rate is 8% compounded annually. What should the monthly payment be on the $100,000 loan?

The solution given is 1198.58


But if I use this formula I got 1213.28 instead

Can someone help me to understand how to calcualte the monthly payment in this case and why we can't use the second formula

2 Answers 2


The difference is how you interpret "8% compounded annually" In reality, loan rates are quoted as "annual", meaning the actual monthly interest rate is n/12 (r/n in your formula). Most loan payment formulas (including the one you used) make this assumption.

When instead you say "compounded annually", you need to find the monthly rate that corresponds to 8% annually when you apply it to a monthly payment (compounding it 12 times). So the difference is a monthly rate of 0.6434% versus 0.66667%


As D Stanley comments, the problem is with the rate.

"8% compounded annually" is a 'nominal' rate, as identified by the statement of a compounding period.

A 'nominal rate compounded annually' is exactly the same as an effective annual rate. However, to obtain a nominal annual rate compounded monthly ( i ) from an effective annual rate ( r ) . . .


as is described on Wikipedia: Effective interest rate calculation.

i.e. with  r = 0.08 effective annual rate
           n = 12
           i = n ((1 + r)^(1/n) - 1) = 0.0772084 compounded monthly

What does this mean? The expression (1 + r)^(1/n) - 1 is the interest per month ( = 0.6434%), calculated from the annual effective rate. And this is simply multiplied by 12 to obtain the 'nominal rate compounded monthly'. It is defined like this so that given a 'nominal rate compounded monthly' the monthly interest can be obtained by simple division by 12. Handy for the days before pocket calculators, as described here: 2008 Federal Reserve memo, i.e. "the true calculation [involving inverse powers] ... was not readily available".

Using this converted rate ( ì ) the payment calculation can be completed.

P = 100000
t = 10


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