Usually people come across various forms of loans in their lifetime, which the mostly commonly seen in the USA are fixed rate loan for automobiles or houses. There is usually a fixed payment each period and by the end of the loan, all payments including interests and principals will be paid in full. However, how is the monthly payment calculated?

If you got a quote from the lender, the quickest way is to use some online calculators for mortgages and pretty much everything is a plug and play. In order to derive the equation on your own, there are also a few approaches. And here I am going to cover some of them. Using a financial calculator, using time value of money and the hard core way, derive the monthly payment using the definition of loan.

# 1. Using Texas Instrument BA II Plus financial calculator

It is pretty easy to use the financial calculator to calculate the monthly payment. I uploaded a Youtube video.

You can also quickly check with the calculator in Google to confirm the calculation was right.

# 2. Time Value of Money

In finance, people like to evaluate the intrinsic value of an asset to be the net present value of all future cashflows. In a 30 year fix rate mortgage with equal monthly payment, we can assume that the monthly payment has the value pmt. And for the payment at period N, the net present value of pmt should be discounted using the internal rate of return which should be period interest rate, monthly interest rate.

For example

NPV(pmt_12) = pmt / (1+r)^(12) where r = Annual Interest Rate / 12

In this case the net present value of all future cashflows will be sum of all monthly payments.

# 3. Monthly payment derivation

put the numbers into the equation, we got the right result.