Amortized Loans

Overview


An amortizing loan is structured so that the payments on the loan are all equal. This means that each payment will include a payment for both interest, and a portion of the principal. Mortgages are a common example of an amortized loan.

Beacause each payment contains a payment for principal, the interest due each period declines. Becuase the payment is a constant fixed amount each period, this means that the payment will include an ever increasing amount of principal until the principal reaches zero. That is, initially, interest comprises a large amount of the payments, with a small principal payments. Payments later in the stream will be predominantly principal.

Calculating the Monthly Payment


Given a loan amount (the principal) and a desired number of periods (in this case, periods is the number of months) and a given interest rate, {% r = rate/12 %}
{% Payment = Principal \times (1 + r)^{periods} \times [r/((1+r)^{periods} -1)] %}
The first term, {% Principal \times (1 + r)^{periods} %}, is the future value of principal at the maturity.

Implementation


The amortization formula above can be implemented in a straightforward manner as 


let payment = principal * Math.pow(1+r, periods) * r /(Math.pow(1+r, periods)-1)
Try it!

Remaining Principal


One can verify the veracity of this formula by iterating over the periods and calculating the appropriate values. This sample code iterates through each period and adds the computed interest and subtracts the payment from the principal. The final value will be zero. (In this case, the outputted number is not exactly zero, but very close due to the vagaries of floating point arithmetic) 


let principal = 100;
let r = 0.1/12;
let periods = 30*12;
let payment = principal * Math.pow(1+r, periods) * r /(Math.pow(1+r, periods)-1);

for(let i=1;i<=periods;i++){
  principal += principal * r;
  principal -= payment;
}
Try it!


The value of the principal at any point prior to maturity can be calculated by stopping the above loop after the desired number of periods.

API


The amortization payment formula and remaining principal calculations is encapsulated in a amortiziation module, which aids in code readability.

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