Investment Science: Perpetual Annuity - How to compute your monthly loan payment

Friday, August 27, 2010

In this post on Investment Science, we will look into the perpetual annuity formula with a simple example.

What is an annuity? In simple words it's a stream of fixed payments over a fixed period of time, the best example is mortgage, consider a car loan or a home loan you may have taken, how to derive the monthly payments for your loan, how much interest you pay for over the period of your loan, how much interest your pay every month, how much balance principal you pay in your monthly loan  payment? Which is beneficial, paying additional mortgage for short term or paying less mortgage for long term? Questions galore. In the next few posts we will be exactly looking into these.

Keep in mind that learning the mathematics of your mortgage payment will help you make informed decisions and to start with that, just remember this simple, yet powerful one, the annuity formula.

The present value (P) of an annuity which pays an amount A every individual period (an year) for 'n' years at an yearly interest rate 'r' is given by


Note: If the number of periods per year is > 1, say 'm' periods per year (m = 12 for mortgages since mortgages are paid monthly), the above formula for a total of n years becomes


This simple, yet powerful formula forms the basis for annuity payments, we will look a practical example.

Example:

Suppose you have taken a car loan for $15000 which is to be paid every month over 5 years at an interest rate of 5%, then how much you need to pay every month.

Here

P = $15000
m = 12 (Monthly payment)
Monthly interest rate = r/m = 5%/12 = 0.05/12 = 0.004166
Number of years the loan is to be paid, n = 5
Number of periods the loan is to be paid = n * m = 5 * 12 = 60

Therefore


A = $283.063

Therefore over the period of 5 years, the interest you would have paid = A * (n * m) - P = 283.063 * 60 - 15000 = $1983.78042


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