Why Study Math? – The Mathematics of Finance – Calculating Your Mortgage Payment

In the first two parts of this series we discussed how compound interest is computed and the effects of various compounding on your net return. Here we discuss how that dreaded of all dreaded payments is calculated. What is it?–yes, you got it, that death pledge of a debt–the mortgage. You’ll want to read this.

If you don’t already know, mortgage derives from two French words which mean “death pledge.” When you consider all the foreclosures that are occurring right now after the sub-prime bust, the etymology of the word rings pretty true to life. In this article, we are going to discuss the way to calculate your mortgage payment based on the specified term and interest rate. You need to understand compound interest and nominal rates of interest, so if you have not mastered these two topics from my articles “The Mathematics of Finance” Parts I & II, go and read those before you try to tackle this one.

A mortgage is actually a form of an annuity: a contract in which one party, in exchange for a lump sum of money, promises to make a stream of payments over a certain period of time. When a bank gives you a mortgage, the bank is giving you a sum of money with which to purchase your home, in return for your series of payments, which are generally paid every month over a period of thirty years. With the knowledge from the first two articles, you can calculate this payment quite easily.

Let’s assume that “Frequent Compounding Bank USA,” your friendly local lending institution, grants you a thirty year mortgage for $100,000 at 6% interest. The way the bank figures the compounding on mortgages is by using the monthly nominal rate. Thus at 6%, the nominal rate is 6%/12 or 0.005. The way we obtain the monthly payment is by using the formula that states that the monthly payment P times the annuity factor (which we will call an) is equal to the amount borrowed A. Using 6% and $100,000, this formula translates to P*an = A, or P*an = $100,000.

Solving for P, we have P = A/an. All we need to know now is what an is equal to. To find an, we introduce another factor, called the discount factor, and we denote this by v. V is equal to the reciprocal of one plus the nominal interest rate. Mathematically v = 1/(1+i), where i =.005. The annuity factor an is expressed as follows: an = (1 – v^n)/i, where n is the number of months.

Let’s take our example of a $100,000 thirty year mortgage at 6% and calculate our payment P. Note that 30 years is equal to 30*12 or 360 months. V = 1/(1 + i) or 1/1.005. Thus v is equal to 0.99502, to four decimal places. We can now find our annuity factor an. Thus an = (1 – v^n)/i, or (1 -.99502^360)/.005. When we enter this into our calculator we get an = 166.85. We can now calculate P as P = $100,000/166.85, or P = $599.35.

Yes, that’s all there is to calculating that dreaded of all dreaded monthly payments. Just remember. This death pledge is not a death pledge unless you make it one. Don’t.

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