# 3. Math of House Buying

by M. Bourne

### Disclaimer

This discussion is simplified so we don’t get lost in complications. Also, interest rates are changing all the time - check your local banks for latest rates.

There is no need to really use the formulas in this section. You can easily use Excel to calculate the values or you can use the many on-line calculators to find the values.

This site does not contain investment advice.

Don’t miss interactive mortgage calculators on this page.

Tom (see the chapter intro) wants to buy a house sometime soon. He knows that real estate is a reasonably safe way to build wealth. It is slow, but generally low risk.

He will come across a bewildering choice of **mortgages**.
A mortgage is just a fancy word for the process of borrowing
money for a house.

The bank actually buys the house and keeps the title deed. You agree to pay off the borrowed amount each month, usually over a period of 20 to 30 years. When you have finally paid it off, you get the title deed, meaning you now fully own the house.

If you stop paying for any reason, the bank has the right to sell the house. After all, they own most of it.

## Mortgage Example

Tom’s house will cost `$300,000`. He needs to pay a **deposit**
of `10%` and will pay the remaining `90%` over `30` years at `8%` per
annum.

So he will need to pay a deposit of `10%` of `300,000 = $30,000`.

The remaining amount he owes is `$270,000`.

### Monthly Payments

The formula for the amount Tom has to pay each month is

`A=(Lxxr)/(1-(1+r)^-n)`

where

A= amount to pay each month

L= loan amount (orprincipal)

r= interest rate (per year as a decimal - or divide by `12` to get the rate per month)

n= number of payments

(This formula is based on the Sum of a Geometric Progression.)

So for our case, we have:

`L = 270,000`

`r = 8% -: 12 = 0.0066667`

`n = 30 xx 12 = 360`

So the amount is

`text(Payment)=(Lxxr)/(1-(1+r)^-n)`

`=(270000xx0.08/12)/(1-(1+0.08/12)^-360)`

`=$1981.16`

So Tom needs to find just under $2000 per month to buy his house. Of course, he won’t be paying rent so the extra amount is not too bad, he figures.

## Back of the Envelope Calculation

A reader asked if there is a reasonable approximation that we can use for this formula when we want to do "back of the envelope" calculations. It turns out there is.

("Back of the envelope" is an expression meaning we can work it out quickly and easily on a scrap of paper.)

Most mortgages are for less than 30 years. The following chart illustrates the value of the denominator for the above expression, for interest rates of `6%` to `14%` up to 30 years.

For example, the graph for lowest curve, `6%`, is of the following form, where *n* is the number of years:

`1 - (1 + 0.06/12)^(12n)`

We can use this graph as follows.

If the mortgage is `25` years and the interest rate is `10%`, the graph tells us the denominator value is close to `0.91`.

So a `$100,000` loan would have a monthly payment of about

`(100,000xx0.1/12)/0.91=916`

A check in the actual formula yields a value of `$908.70`. So the approximation is not bad.

## Interactive Mortgage Payments Calculator

You can calculate any repayment here. Change any of the values for principal, interest or period and then click "Calculate". You can also see a graph of the **equity** (the amount you own) that has been built up in the property over time.

If you **move your cursor** over any of the graph bars that appear, a tooltip will tell you how much is still owing, and how much equity you have in the house.

The amounts show are for the **beginning of the year** in each case, so at the beginning of Year 1, you owe the full amount and have no equity, while at the beginning of Year 31 (in the example given), you have paid it all off and now you own the house outright.

**NOTE:** This interactive assumes **zero house price inflation**. If you are lucky, your house will increase in value and will be worth more than you paid for it. But that doesn't always happen.

Principal = $ (min 50000, max 10 million)

Interest p.a. = % (min 0.01, max 100)

Period = years (min 5, max 35)

Year

Copyright © 2013 www.intmath.com

**Note: ** These calculations do not take into account the bank fees and we are assuming a fixed rate of interest for the whole period of the loan. Actual amounts charged by lenders are sure to be higher than this.

### Six Months Later

Tom has been paying off his pride and joy for 6 months. He has paid a total of `$1981.16 xx 6` ` = $11886.96`.

He gets the first statement from the bank and expects to see a reasonable dent in the amount he owes. He is shocked to find that he still owes `$268,894.74`. How is this possible? He has paid `$12000` but only around `$1000` has come off the balance.

The formula for the balance is:

`text(Balance)=(L[1-(1+r)^(p-n)])/(1-(1+r)^-n)`

where

L= the loan amount

r= interest rate per month as a decimal

p= number of payments already made

n= total number of payments to be made

In our example, the balance will be:

`text(Balance)=(L[1-(1+r)^(p-n)])/(1-(1+r)^-n)`

`=(270000[1-(1+0.08/12)^(6-360)])/(1-(1+0.08/12)^-360)`

`=$268\ 894.74`

With mortgages, the amount you are paying early in the loan period is mostly interest, and very little is coming off the principal. Towards the end of the loan, the interest amount is less and the principal starts to disappear more quickly.

Amount ($) still owing after *p* months.

The mistake a lot of people make is to sell the house quickly. The average mortgage is only 7 years. They own very little of the house by then because they have mostly been paying interest to the bank. If house prices have gone up a lot, they are ahead. But if house prices level off, or decline, then you lose a lot of money.

## Reader Question

Reader David asked:

Is it possible to derive an equation for when the interest part of the payment for a loan exactly equals the principal part of the payment?

There is, David, and here's how to do it.

## Money Maths Lesson Plan Suggestion - House Buying

Simulate a house buying scenario in your district. Use actual advertisements for houses and for housing loans. Get students to find the best loan deals. Which is best - fixed or variable interest? What will they pay in total for the house? What will it be worth at the end of the loan period? How much will they have paid after 10 years and how much will they own by then?

#### Footnote

It may not always be best to buy a house compared to renting. In Japan, house prices have still not recovered to where they were 16 years ago. Sure, interest rates are low there, but **negative equity** has been a huge problem for years. (This is when the amount that you owe on a house is more than what the house is worth.)

House prices, Japan. Index = 100 in 1975.

This means that house prices more than doubled at the peak in 1991 and have dropped back to 1975 levels since. In 2005/2006 there has been a modest increase.

Interestingly, house prices in US, Australia, UK, and most industrialised economies have shot up in recent years due to low interest rates. One wonders what the future holds for house prices in these countries...

If rents are low in your area, it may be better to rent a place and invest (not **spend**) the difference.

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