Video Captions: Algebra Applications: Data Analysis
[Music]
Title: Algebra Applications: Data Analysis
Title: The Mortgage Crisis
Narrator: In 2008 many banks in the United States
began to fail, and many more stopped lending money.
The reason was that many homeowners were defaulting on
their mortgages.
Banks were not getting paid for the money they lent, and
as a result they stopped lending money.
With less money flowing through the system the US
economy went into a recession.
The stock market plunged and hundreds of thousands of jobs
were lost.
The root of the economic crisis was the problem
around mortgages.
A mortgage is a loan.
A bank lends you the money to buy a house and charges you
interest on the loan.
As you pay back the mortgage you not only pay back the
amount that you borrowed, you bay back the interest on
the loan.
Mortgages are as old as banks themselves.
People have been buying houses this way for centuries and
banks have been successfully lending money this way.
So what happened in 2008 with this successful system?
Why were so many people suddenly not able to pay
their mortgages?
We will investigate this problem first by analyzing
what a mortgage is, then we will look at different types
of mortgages, especially the ones most responsible for the
mortgage crisis.
In particular, we will look at so-called sub prime mortgages
to see what made these types of loans so toxic to
the economy.
So as a first step let us start with an investigation of
what a mortgage is.
[Music}
Narrator: A mortgage is a loan to purchase a house.
Mortgages are long-term obligations.
Most mortgages are loaned for a thirty-year period.
Every mortgage consists of several key pieces of data
:the loan amount, the interest rate, and the number of
payment periods.
The loan amount is the amount of money the bank lends you to
purchase the house.
This is the money you will pay back to the bank.
The interest rate is the additional amount of money you
will pay to the bank for borrowing the money.
The interest is what the bank earns for lending you
the money.
The number of payment periods is the number of months the
bank gives you to pay back the loan.
Let's look at some examples.
This table shows a thirty-year loan for the amount of one
hundred thousand dollars.
This is the loan amount.
The interest that the bank is charging you is five point
two-five percent.
The payment period is three hundred and sixty months, or
thirty years.
Using these three pieces of data you can construct an
amortization table, which is the schedule of payments to
repay the loan.
Let's use the TI-Nspire to do this.
Turn on the TI-Nspire.
Create a new document.
Decide if you want to save your document.
Create a spreadsheet window.
Input the following text labels into cells A1, A2, A3,
and A4.
Then input the following values in the corresponding
cells of column B.
We will be calculating the monthly mortgage payment for a
thirty-year mortgage of one hundred thousand dollars paid
at an interest rate of five point two five percent.
In cell B4 you will be using one of the Nspire's finance
functions to calculate the monthly payment.
The time value of money function takes the value from
cells B1 to B3 to calculate the monthly mortgage payment.
Input the first payment of the function as shown.
The time value function evaluates six numbers.
The first is the total number of payments.
So after the left parenthesis input cell reference B3.
The next value is the interest rate.
So after the comma input cell reference B2.
The next value is the loan amount, so after the comma
input cell reference B1.
The next reference is the future value of the mortgage.
Since you intend to pay the entire mortgage the future
value is zero.
So after the comma input zero.
The next two values are the number of payments per year
and the number of times the interest on the loan
is compounded.
In both cases assuming monthly amount.
So after the comma input twelve two times.
Complete the formula and press ENTER.
The time value function shows that the monthly mortgage
amount is about five hundred and fifty two dollars per
month for three hundred and sixty months.
To find out how much money you actually pay input the
following into cells A5 and B5.
You'll see that a one hundred thousand dollar loan paid over
thirty years cost nearly two hundred thousand dollars at a
loan interest rate of five point two five percent.
Suppose that the interest rate went down to five percent.
How does this affect the loan?
Change the value in cell B2.
Press ENTER.
A small change in the interest rate will result in several
thousand dollars worth of savings over the life of
the loan.
Now you can see why even minor changes in the interest rate
are important.
And this provides the first clue about what may have
occurred in the mortgage crisis.
But there's more.
Although every month the mortgage is the same
throughout the life of the loan, the amount that goes
towards paying back the one hundred thousand dollars, and
the amount that goes toward paying the interest on the
loan varies from month to month.
To see how this month-to-month payback works lets create an
amortization table.
What we want to keep track of are three things: the monthly
balance- how much is still left to pay on the loan, the
amount paid toward the principle, which the one
hundred thousand dollars borrowed, and the amount paid
toward interest.
Use the nav pad to restore the five point two five percent
interest in cell B2.
Move the cursor to the top of column C.
Input the column heading "balance".
The cursor is now in the gray cell where you can add
formulas to generate data.
We want to generate a sequence of numbers that reflects the
changes in the loan balance.
Since every month you pay five hundred and fifty-two dollars
you reduce the loan amount by this payment, but every month
you need to pay interest on the balance of the loan so we
need a formula that reflects this.
Let P represent the previous month's balance.
Our equation then changes to this.
The interest rate is divided by twelve because we are
calculating the monthly payment.
The five hundred and fifty-two represents the
mortgage payment.
Using the interest rate of five point two five percent
the equation changes to this.
We can generate the sequence of numbers representing the
loan balance during the three hundred and sixty months of
the repayment.
Press control and MENU and select generate sequence.
At the dialog box input this formula.
The term u(n-1) represents the previous term
in the sequence.
The cell references B2 and B4 are for the interest rate and
the monthly payment.
Press TAB to go to the next cell.
Here you add the initial value, which is the original
amount of the loan, or one hundred thousand dollars.
Press TAB and set the number of terms in the sequence as
three hundred and sixty, which is the number of
loan payments.
Keep pressing TAB to get to the OK button and press ENTER.
If you scroll down you will see the complete list of three
hundred and sixty terms representing the monthly
balance for the mortgage.
Now move the cursor so that it is at the very top of
column D.
Add this label.
Press ENTER.
The cursor is now on a cell there you can write a formula
to generate data.
You want to generate the interest payment made
each month.
This means multiplying the current loan balance by the
monthly interest.
Input this formula and press ENTER.
This generates a sequence of three hundred and sixty terms
that represent the monthly interest payment.
Now move the cursor so that it is at the very top of
column D.
Add this label.
Press ENTER.
This column will track the monthly payment on the
principle of the loan, the one hundred thousand dollars.
Input this formula and press ENTER.
We now have a complete amortization table for
the loan.
The first thing to notice is that at the beginning most of
your mortgage payment goes toward interest, not
the principle.
This becomes clear by looking at a SCATTER PLOT of columns
D and E.
Press HOME to create a graph window.
Press MENU, and under GRAPH TYPE select SCATTER PLOT.
Move the cursor to the drop-down MENU for x.
Select PRINCIPLE.
Press TAB to highlight the y drop-down MENU.
Select INTEREST.
To see the data press MENU and under the WINDOW option select
ZOOM DATA.
You'll see a scatter plot of interest versus
principle payments.
At the start of the loan most of your payment is
toward interest.
It's only until the two hundred and third payment, or
nearly seventeen years, that most of the monthly mortgage
payment goes toward the principle.
If you input a higher interest rate you'll see that it takes
even longer before you reach the point where most of the
payment goes toward the principle.
As you begin to pay down the principal you gain what is
called equity on the house.
Equity is the portion of the loan that you have paid back.
Think of it as the portion of the house that you own.
To see how equity grows let's use the cumulative
sum formula.
GO to the top of column F and add the label "equity".
Use the cumulative sum formula for column F.
Scroll down the equity column.
You'll see that it takes two hundred and forty-five
payments before you have made half the principle.
In other words it will take twenty-one years before half
of the mortgage is fully paid.
During this time the value of the house will likely go up.
Suppose that this one hundred thousand dollar house gains
five percent in value every year.
The total equity is the sum of the increased value of the
house and the total amount of principle you have paid back.
So buying a house is a long-term commitment.
When a bank lends money for a mortgage the expectation is
that the person buying the house has the commitment to
pay the loan.
Even though it takes a long time to pay it back there is
also the benefit of having the house gain in value so that
the total equity on the house continues to increase.
During the mortgage crisis of 2008 neither of these
assumptions turned out to be true.
Many people stopped paying their mortgage and the value
of houses began to go down.
Now that you know how to calculate a mortgage we can
begin to analyze what went wrong during the
mortgage crisis.
Be sure to save the amortization template
you created.
Call the file "amort".
[Music]
Narrator: Two families are buying a house.
Each takes out a thirty-year mortgage for one hundred
thousand dollars.
One family has a monthly mortgage payment of five
hundred and seven dollars, but the other family has a
mortgage payment of six hundred and
thirty-two dollars.
Clearly one family is paying back the loan with a higher
interest rate than the other family.
In fact, the first family has a four point five percent
interest loan while the other family has a six point five
percent interest loan.
Why does this happen?
The reason that different people pay different mortgage
amounts has to do with their credit score.
A credit score is a number that banks use to determine
the interest you will be charged for a loan.
There are several different credit scores, but one that is
used a lot is the FICO credit score, which has a range from
three hundred to eight hundred and fifty.
The higher a person's FICO score the lower the interest
rate they will be charged.
The lower a person's FICO score the higher the interest
rate they will be charged.
A FICO score determines how much of a credit risk you are.
This table shows the national distribution of FICO scores.
In particular, look at the third column of data.
As your FICO score goes up the less likely you are to default
on your loan.
As you can see from the graph more than fifty percent of
people have a FICO score greater than seven hundred.
These people have a combined delinquency rate of
eight percent.
As a result banks will entice these borrowers with a lower
interest rate.
This box and whisker plot shows that the median FICO
score is seven hundred and twenty-three.
This means that half of all borrowers have a FICO score
greater than seven hundred and twenty-three, and half have a
FICO score less than seven hundred and twenty-three.
Those in the upper range have a very small delinquency rate.
There is roughly twenty-five percent of people who have
credit scores between six hundred and seven hundred.
These people are a higher credit risk and banks will
charge them a higher interest rate.
One type of mortgage that people in this category get is
called a sub prime mortgage.
This table shows the different interest rates charged based
on FICO scores.
Note that interest rates vary from time to time, but what is
constant is that the higher FICO scores lead to a lover
interest rate.
Let's compare the mortgage payments for two different
FICO scores.
Open your amortization table template.
Move your cursor to cell B2.
Suppose that one person's FICO score is eight hundred.
According to the table their interest rate would be four
point five five three percent.
Input this value into cell B2.
Look at the value in cell B4.
This is the monthly mortgage payment.
Cell B5 shows the total amount that this person will pay into
the loan.
Scroll down the spreadsheet to see when the payment on the
principal exceeds that on the interest.
It takes one hundred and seventy-eight months, and at
that point the person has paid back about one-third of the
loan amount.
Now compare this to someone with a score of six fifty.
According to the table their interest rate would be five
point five nine six percent.
Input this value into cell B2.
This table shows how much more difficult it is for someone
with a sub prime mortgage.
Not only do they have a higher monthly payment, but they gain
equity on the house much more slowly than someone with a
good credit rating.
On top of that, the person with the lower FICO score has
a higher probability of defaulting on the loan.
Here is the table again but with additional column added
for the probability of a delinquency.
This means that for every one hundred mortgages to people
with a credit score of eight hundred only one percent
will default.
However, for every one hundred mortgages to people with a
credit score of six hundred and fifty, fifteen percent
will default.
But a default on a thirty-year loan can happen at any time
during the three hundred and sixty month repayment period.
We can run a simulation to see what the default on these
loans could look like.
Return to the spreadsheet tablet and go to the very top
of column G.
Add this column heading.
Press ENTER.
Make sure the cursor is on the cell above G1.
We will conduct a probability experiment using what we know
about lenders with this credit score.
Basically we want to generate random numbers from one
to one hundred.
Input this formula.
If you press control and R the formula generates a new
random number.
Scroll down and you will see that random numbers from one
to one hundred have been generated for all three
hundred and sixty payment periods.
If a number is less than or equal to fifteen then we
consider that a defaulted loan.
Depending on when it occurs the loan balance owed to the
bank also varies.
Move the cursor to the top of column H.
Add this label.
Press ENTER.
Make sure the cursor is above cell H1.
Our simulation will work like this.
If a value in column G is fifteen or less then that is a
defaulted loan and we'll treat the loan balance as money that
the bank has potentially lost.
Input this formula.
Press ENTER.
Scroll down column H and you will see all the defaulted
mortgages, each with a different loan value.
Let's add them up.
Go to cell J1 and input this formula.
Press ENTER.
Since these are random numbers your results will vary but the
default amount will likely be in the range of two to four
million dollars.
As the number of sub prime mortgages increase so does the
amount of bad debt.
Compare these results to the situation with someone with a
credit score of eight hundred.
You'll need to change the following spreadsheet cells.
Change the value in the B2 to four point five five three.
Change the formula at the top of column H to the one shown
in the table.
After making the changes press control R and re-run
the simulation.
You'll see that the amount of loan defaults in terms of
dollar amounts is much smaller.
We are now closer to understanding what caused the
mortgage crisis of 2008.
One key factor was the number of loan defaults and this was
due to an increasing number of sub prime loans.
But what other factors could have made the problem worse?
What caused an increase in loan defaults?
For that we need to look at another type of mortgage known
as an adjustable rate mortgage.
♪
Narrator: The mortgage crisis of 2008 was due in part
to an increasing number of sub prime mortgages.
Because these types of mortgages are often given to
those with low credit scores, one consequence of this is an
increase in the number of loan defaults.
Defaulting on a loan means that you stop paying it back.
When a homeowner defaults on a loan the bank takes ownership
of the house.
So before resorting to a loan default a homeowner has
the option of selling their house.
This is almost always the better option since defaulting
on a loan will decrease your credit score even more, making
it all the more difficult to get future loans.
One of the reasons that mortgage defaults increased is
due to the combination of two things: adjustable rate
mortgages and the decrease in the value of real estate.
Let's take a look at adjustable rate mortgages.
We saw that those with a low credit score often pay a
higher interest rate, but this also results in a much higher
monthly higher payment.
All to often these higher payments prevent people from
taking out these loans.
One type of mortgage that helps with this problem is an
adjustable rate mortgage.
This is how they work.
For the first three to five years of the mortgage the
homeowner pays the loan at a lower rate.
The lower rate results in a lower monthly payment.
But after this period of time the interest rate changes to a
higher rate.
The idea is that during this early period the homeowner has
an opportunity to improve his or her credit score.
When this happens then they can refinance their mortgage
to take advantage of lower interest rates.
Otherwise the homeowner will be paying a much higher
mortgage payment every month.
Let's use the TI-Nspire to analyze how the
payments change.
Open the amortization template that you previously created.
Suppose that a homeowner takes out a two hundred thousand
adjustable-rate mortgage.
For the first five years the rate will be four point five
percent, but in the sixth year the rate increases to seven
point five percent.
Change cell B1 to two hundred thousand.
Press ENTER.
Then change cell B2 to four point five.
Press ENTER again.
If a homeowner were to pay this mortgage for thirty
years, then the monthly payment would be one thousand
thirteen dollars and thirty-seven cents
every month.
But keep in mind that this is an adjustable-rate mortgage
that has the rate for five years.
Since five years is equal to sixty months scroll down to
find the sixtieth mortgage payment.
The value in cell C60, one hundred and eighty-two
thousand, six hundred and forty-four is the
loan balance.
This is how much money you still owe the bank on the
original two hundred thousand dollar loan.
Make a note of this number.
From the sixty-first payment on your interest rate changes
to seven point five percent.
Go back to cell B1.
Input the value from cell C60 and press ENTER.
This represents the new mortgage amount.
Input the value of the new interest rate.
You're almost done.
You now need to modify the sequence because it is set up
to calculate three hundred and sixty payments, but after five
years you have three hundred payments left.
Go to the formula line of column C.
Select OK and press ENTER.
You will be asked if you want to replace the data currently
in the column, press ENTER.
The new mortgage payment is one thousand three hundred
forty-nine dollars and seventy-two cents.
So although the mortgage interest went up three
percent, the monthly payment goes up by over
thirty percent.
The extra three hundred and thirty-six dollars a month
turns into more than forty-one thousand dollars over the
entire life of the loan.
So adjustable rate mortgages are good during low interest
periods, but these loans could become very difficult to pay
back during the periods of higher interest.
A common strategy for avoiding the higher monthly payments
was for homeowners to sell their house.
This is a good strategy if the price of the house
has increased.
This graph shows the median price of houses in Los Angeles
County from 1990 to 2008.
Buying a house during the 1990's and selling it before
2006 would have resulted in a profit.
But as you can see from the graph selling a house after
2006 would have resulted in losing money.
Let's take a look at the situation in more detail.
This data table shows the home prices from the graph.
Suppose somebody takes out an adjustable rate mortgage for a
house in 2004 and pays the median price.
Furthermore, suppose that the interest for the first five
years is four percent, and after that the interest rate
adjusts upward to seven point five percent.
Let's investigate.
Go to cell B1 and input the mortgage amount and then input
the interest for the first five years in cell B2.
Go to the formula in the formula row of column C and
change the mortgage period to three hundred and
sixty months.
The monthly payment is two thousand one hundred and
thirty one dollars and nine cents.
Scroll down to see the loan balance after sixty months.
The loan balance is four hundred and four thousand,
five hundred and twenty-two.
Go back to cells B1 and B2 and adjust the loan balance and
the interest rate, then adjust the sequence formula.
In 2008 the new monthly payment would be two thousand
nine hundred and eighty-nine dollars and
thirty-eight cents.
This represents a dramatic increase of about forty
percent in the monthly mortgage payment.
To make matters worse, suppose this homeowner tried selling
the house.
According to the chart the homeowner would get three
hundred and forty thousand dollars.
But according to the amortization table the amount
owed on the loan would be roughly four hundred
thousand dollars.
This means that the owner would still be sixty thousand
dollars in debt.
Between paying forty percent more every month for a
mortgage payment, or selling the house at a loss, the
homeowner is in a difficult position.
Many homeowners in this situation would choose to stop
the mortgage and default on the loan.
Throughout 2006 and 2008 many homeowners found themselves in
this very position, owing more on the house than it was
worth, otherwise known as being under water.
Let's take a step back and see how this affected the
overall economy.
A bank loans money for you to buy a house and in turn you
give money back to the bank.
This cycle of money allows the bank to lend to
other homeowners.
The banks also sell mortgages to larger banks and government
institutions, further allowing banks to lend more money
for mortgages.
The flow of money from homeowners to banking
institution makes up a lot of economic activity.
When foreclosures accelerated from 2006 to 2008 the flow of
money slowed, and in some cases stopped.
This created a credit freeze because not enough money was
flowing through the system.
The reason this became a global crisis is because many
of those mortgages were repackaged and sold to
International banks, and thus a period of great financial
unrest began.
[Music]