The biggest purchase most people make in their lifetime is a house. It is rare for someone to
be able to pay cash for their home; the vast majority will need to borrow most of the money.
Banks make many of these loans, which are called *mortgages*. A mortgage is typically
a 30-year loan, and in April 2009 could have an interest rate below 6 percent.

Banks are willing to loan you the money for your house because the house itself is used as
*collateral* to guarantee the loan. If you are unable to make the monthly mortgage
payments, the bank can *foreclose* on the mortgage, which means they sell your house to
pay off the rest of the loan. Obviously you will need to find somewhere else to live if that
unfortunate occurrence happens to you.

The best way to prevent a foreclosure is to not buy a house that you can't comfortably afford. This may sound like overly simple advice, but the housing crisis that started at the end of 2008 can be partially blamed on people buying houses they couldn't afford. On the other hand, the banks allowed people to get into this situation by not verifying their customers' ability to make their payments. And the banks were encouraged by the U.S. Congress to make these bad loans to increase home ownership. There is plenty of blame to go around. But we're here to learn the math behind mortgages so that you can make smart decisions and not end up as a statistic.

Let's start off by defining the terms of our mortgage. The house we want to purchase costs
$250,000. The bank wants us to show our commitment to paying off the loan, so we are required
to pay 20% of the purchase price as a *down payment*. After paying the bank our down
payment of $50,000, we owe $200,000 on the house, and the 30-year mortgage has an interest
rate of 6 percent. Interest will accrue monthly, which works out to 1/2 percent per month
(6 percent divided by 12 months). Our goal is to determine the monthly payment required to
fully pay off the loan in 30 years.

The first observation we can make is that in the first month of the loan, the amount of interest accrued is:

Interest | = | $200,000 × 0.005 |

= | $1,000 |

If our monthly payment was $1,000, that amount would cover just the interest and we would still owe the full $200,000 going into the second month. The interest due in the second month would also be $1,000; after making the second $1,000 mortgage payment, we would again still owe the full amount of the loan. This can continue indefinitely, with the result being that we never gain any ownership of the house (besides the down payment).

Such a loan is called an interest-only loan, and we would need to make a *balloon
payment* equal to the full loan amount at its conclusion. Since we want to gradually
earn ownership in our home, and not have a huge balloon payment to look forward to, we
will need to pay more each month than this lower bound of $1,000.

If we ignore the interest accrual for a moment and just look at paying off the principal over a time period of 30 years, we would need to make 360 monthly payments of:

Payment | = | $200,000 ÷ 360 |

= | $555.56 |

You might guess that our monthly payment would just be the $1,000 interest payment plus this $555.56 payment toward the principal. That would actually be too high, since over time the principal is decreasing, so the amount of interest accruing each month is also decreasing. The good news is if we can afford $1,555.56 per month, then we can definitely afford to buy the house, since it is an upper limit to our payment.

To figure out the exact payment, we need to pull out our trusty compound interest equation. We'll do something a little tricky to get there, but we'll end up with a compact equation that determines our monthly mortgage payment. So try to follow along if you can, but don't worry if it's confusing.

To start out, let's imagine that we make no payments at all during the 360 months of the mortgage to see how much we would owe:

Obligation | = | $200,000 × 1.005^{360} |

= | $1,204,515 |

Note that this result does not mean that we end up paying $1.2 million for the house; we will see how much is actually paid shortly.

Each month we will make a payment, P, toward the purchase of the house. The first of these payments will accrue interest for 359 months to determine how much of the Obligation above is canceled. The second payment is the same amount, P, but it accrues interest for only 358 months. Each subsequent payment accrues interest for one fewer month than the previous payment, so our payments have an accrued value of:

Payments | = | P × 1.005^{359} + P × 1.005^{358} + ... +
P × 1.005 + P |

= | P × (1.005^{359} + 1.005^{358} + ... +
1.005^{1} + 1.005^{0}) |

To determine our payment, P, simply assign the Payments equal to the Obligation and solve the equation. A useful formula that greatly helps compute the sum of powers above is:

1.005^{359} + 1.005^{358} + ... +
1.005^{1} + 1.005^{0} |
= | 1.005^{360} − 1 |

1.005 − 1 |

See the discussion of the sum of powers if you are curious
how this formula was derived.

Our monthly mortgage payment is therefore:

P | = | $200,000 × 1.005^{360} × |
1.005 − 1 |

1.005^{360} − 1 |
|||

= | $1,199.10 |

Since we make 360 equal payments over the life of the loan, the total amount of money we pay is $1199.10 × 360 = $431,676, more than double the cost of the house. But in the end we own a home that we had use of for 30 years, so it is a good deal for us as well as the bank.

I promised a compact equation, but since it's a bit messy, I'll skip the derivation; it follows directly from the equation for P above. Surely you shouldn't try to memorize it; just return here in the future if you need to refer to it:

Payment | = | Loan × | r | ||

12 | |||||

1 − (1 + | r | )^{−360} |
|||

12 |

Using the example above, Loan would be set to $200,000 and r would be 0.06 for an interest rate of 6 percent. The form below will calculate your monthly payment based on the price of the house, down payment, interest rate, and duration of the loan.

Please send comments and suggestions to:
mikestoolbox@pobox.com

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