Introduction to Mortgage Mathematics and Mortgage Backed Securities

In this lesson we were introduced to basic mortgage mathematics and mortgage backed securities and the mortgage market.

The mortgage market accounted for 23.3% of the total outstanding amount of US bonds in Q3 2012.  These markets are huge and played a huge role in the financial crisis of 2008 and 2009.
MBSs are a particular class of asset-backed securities.  These are securities backed by underlying pools of securities, such as mortgages, credit card receivable,s auto loans, student loans, etc.  The process by which ABS are created is called "securitization."

Here is an image of how securitization works.  There are 10,000 mortgages and we combine them into one pool.  There are multiple tranches in the pool based on the risk characteristics of the loans.

There are many types of mortgages:

• level-payment mortgages
• adjustable-rate mortgages and
• balloon mortgages

to name a few.  We will only consider level-payment mortgages in these lessons.  A level payment mortgage is a mortgage in which the payment each month is constant over time.

We are going to consider a standard level-payment mortgage with the following assumptions:

• The initial mortgage price M(0) = M
• We assume equal periodic payments of $B
• The coupon rate  is c per period
• There are a total of n repayment periods
• After n payments, the mortgage principal and interest have all been paid (the mortgage is fully amortizing)

This means some of the payment, B, is principal, and some of the payment is interest.

If M(k) denotes the mortgage principal remaining after the k-th period, then

 


where M(n) = 0.
We can iterate this equation from M(0) = (1+c)*M(1)  - B all the way up to M(n-1) = (1+c)*M(n) - B to get the following equation:

but since M(n) = 0, we can plug k=n in this last equation we can say that:

So if we know M, n, and c, we can find the periodic payment amount for the mortg
age.
We can then substitute this B back into the equation above to get:

This tells us the value of the mortgage outstanding as a factor of the initial mortgage amount, the coupon rate, the number of total periods, and the current period.


Now suppose that we want to compute the present value of the mortgage in a deterministic world, with not possibilities of default or prepayments.
Assuming a risk-free interest rate of r per period, we can obtain the fair value of the mortgage as:

Here we are assuming that r is the borrowing interest rate for banks.  In practice, r will not be equal to c.  However,  if r=c, then F(0) = M(0).  In general r < c as banks must charge a higher interest rate so they can account for payment uncertainty, default, prepayment, servicing fees, and profits.

We can also decompose the monthly payment B into the interest component and the principal component.  The interest is:

is the interest that would be due in M(k)
The k-th payment is:

We can use this later to create principal-only and interest-only MBSs.