The Mathematics of Money

434 Chapter 10 Consumer Mathematics

With few exceptions, the amount borrowed with a mortgage loan must

be less than the value of the property. Mortgage lenders typically set

a maximum percentage of a property’s value that can be borrowed. This is

sometimes called the maximum loan to value percentage (or simply loan

to value or LTV). For example, if a lender’s maximum LTV is 95%, the

largest amount that could be borrowed against a $200,000 property would

be (0.95)($200,000) 
 $190,000.

The difference between the value of a property and the amount that is

owed against it is called the homeowner’s equity, or, more simply, just the

equity. In the previous example, if the value of the property is $200,000

and the amount owed on a mortgage against it is $190,000, the home-

owner’s equity would be $200,000  $190,000 
 $10,000. Rather than

state the maximum someone can borrower, lenders sometimes will state

the minimum equity. You can easily relate these two ideas, though, since

the amount borrowed and the equity must add up to 100%. So if the maxi-

mum loan to value is 95%, we could equivalently say that the minimum

equity is 100%  95% 
 5%.

A property’s owner can have more than one mortgage on a given

property. A first mortgage, as the name suggests, is the primary mortgage

loan. A second mortgage is an additional mortgage loan made against a

given property. In the event of foreclosure, the first mortgage lender has

first claim against the property, and the second mortgage lender gets paid

only if there is money left after the first mortgage is satisfied. For this

reason, a second mortgage will usually have a higher interest rate than a

first mortgage, because while the second mortgage is secured, the security

is not as great as that of a first mortgage.

Example 10.2.1 Les and Rhonda own a house worth $194,825. The balance they

owe on their fi rst mortgage is $118,548. They want to take out a second mortgage,

and all of the lenders they have spoken with require a minimum equity of 5%. Find

(a) their equity now, (b) the maximum amount they can borrow with the second

mortgage, and (c) their equity if they borrow the maximum.

(a) The equity is the difference between the property’s value and the amount owed against

it. Since

$194,825  $118,548 
 $76,277

their equity now is $76,277.

(b) Since the minimum equity is 5%, the maximum they can borrow in total is 100%  5% 


95%. So the maximum they can owe in total is

(0.95)($194,825) 
 $185,084

Since they already owe $118,548, the most they could borrow with a second loan is

$185,084  $118,548 
 $66,536.

(c) If they borrow the maximum, they will owe a total of $184,084 and their home equity

would be reduced to

$194,825  $185,084 
 $9,741.

We could also have calculated this by noting that if they borrow the maximum, they will have

5% equity, and so their equity would be:

(0.05)($194,825) 
 $9,741

If Les and Rhonda go ahead and take out the second mortgage, this will greatly reduce

their homeowner’s equity. A second mortgage can be thought of as borrowing against the

equity in the property, and for this reason such loans are often called home equity loans.

While it offers many, many advantages, buying a
home usually involves borrowing a large amount of
money. © Photodisc