# How to calculate monthly payment of a loan

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

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 ) . . .

$i=12((1+r)^{\frac{1}{12}}-1)$

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
``````

$MP=\frac{P(\frac{i}{n})}{1-(1+\frac{i}{n})^{-nt}}=1198.58$