Economics 5-623
Notation
To explore the mathematics of mortgages, we will use the following notation:
M - Original mortgage balance
N - Mortgage maturity, number of periods for which payments are due
R - Mortgage interest rate
INTt - Mortgage interest payments in period t
PAYt - Mortgage payment due in period t
B - Outstanding mortgage balance at the end of period t, B0 = M
The variable t refers to a particular period and t = 1 to N. For level payment mortgages PAYt will be the same for all periods and we can drop the subscript t.
The examples below measure interest rates on an annual basis. For a twenty five year 9% mortgage, R = .09 and N = 25. To use the formulas for mortgages with monthly payments one needs to replace the variable R with R/12 and the variable N with 12N. For example, a twenty five year 9% mortgage would have R = .0075 and N = 300.
Level Payment Mortgages
Mortgage payments, PAY, are determined such that the present value of the N mortgage payments, when discounted at the interest rate R, is equal to the original mortgage balance M. The basic formula for a fully amortized, level payment loan where payments are made at the end of each period is:
A fully amortized loan is one where the periodic payments cover interest and principal. There is no outstanding loan balance following the last payment. See Figures 2 and 3.
One can manipulate the basic equation to derive the expression for PAY shown below. This expression involves four variables, PAY, M, R and N. Given any three variables it can be solved for the fourth. For example, one can use the formula to determine the payment, PAY, necessary to service a loan of M for N periods at interest rate R. Alternatively, one can use the equation to find the maximum loan, M, that can be supported with payments of PAY, based on an interest rate of R for N periods. The equation can also be solved for R given PAY, M and N or it can be solved for N given PAY, M and R. It is not possible to write a simple equation for R. The expression for N involves logarithms.
A fully amortized loan covers interest and principal. For a level payment mortgage the expression above shows that PAY is greater than interest on the original loan balance as the term in brackets is greater than 1.0.
One can also derive equations for the following:
The outstanding loan balance at the end of any period.
t = 1, 2 ... N
The loan balance declines over time as seen in Figure 1. You can verify by substitution that when t = 0, B0 = M and when t = N, BN = 0. Note that at higher interest rates the outstanding loan balance declines more slowly in the early years of a mortgage.
The interest payment for any period, INTt.
t = 1, 2 ... N
Interest payments in period t are proportional to the outstanding loan balance at the beginning of the period which is the same as the outstanding loan balance at end of the previous period. The decline in the outstanding balance over time leads to a decline in interest payments over time.
The principal payment for any period, PRINt.
t = 1, 2 ... N
As principal payments plus interest payments equal the level payment each period, we know that the principal portion of mortgage payments will increase as the interest portion decreases. Figure 2 shows the division between interest and principal over time for a 9% 25 year mortgage.
If the interest rate is higher, mortgage payments must also be higher. It turns out that as interest rates increase the dollars devoted to initial principal repayment decrease. On net the increase in the mortgage payment is less than the increase in interest dollars that must be paid. The following geometric argument may help to explain this point. The simple, undiscounted sum of principal payments must equal the size of the mortgage loan, M. That is the size of the lightly shaded area in Figure 2 must equal M. (The time value of money is shown by the interest component of payments.) Geometrically the rectangle representing total mortgage payments is larger when interest rates increase. Principal payments are still the irregular shaped triangle spread along the upper and right hand sides of the rectangle. It is the size of this funny triangle that equals M. As the mortgage rate increases, the height of the rectangle increases, more of the triangle gets spread along the right hand vertical axis, and the amount of initial payments that is devoted to principal repayment gets smaller. Compare the lightly shaded areas in Figures 2 and 3.
Points and Closing Costs At the time of mortgage origination there are
typically additional costs that borrowers must pay. These include charges for
specific services such as attorney fees, property appraisal, and filing fees.
In addition, borrowers may be asked to pay points, a charge that is computed as
a percentage of the mortgage loan. One point would be equal to 1% of the
mortgage loan, two points 2%. Points are a form of prepaid interest.
Different mortgage interest rates from different lenders are often offset by
differences in points. A lender quoting a low interest rate will typically
charge more points and vice versa. For example see Figure 6 which reports data
from a
The impact of points on the effective mortgage rate depends upon when a mortgage is paid off. If a mortgage is prepaid after one year, 1 point is equivalent to raising the mortgage rate by one percentage point. As a mortgage is held longer the impact of points is less. The exact impact can be calculated by investigating an amended form of the first equation that includes the payment of points at the time of origination, periodic mortgage payments for the estimated holding period, and the payment of the outstanding balance at the end of the holding period. Remembering that PAY is fixed by the contract interest rate, one then solves for the interest rate that equates the present value of points, periodic payments, and the outstanding balance at the end of the holding period to the original loan balance, M.
Adjustable Rate Mortgages
Under ARMs the interest rate may change. Most ARMs in the United States link changes in the mortgage rate to changes in a specified market interest rate. Different ARMs index the mortgage rate to different market interest rates. When the index interest rate changes, mortgage payments are recalculated using the equation for PAY. M would be replaced by the outstanding loan balance and N would be replaced by the remaining number of periods. Typically ARMs have limits on how much the interest rate can increase per adjustment period, so-called brakes, along with an overall upper limit or cap. At times ARMs are offered with a very low initial interest rate that is adjusted after six months or a year regardless of changes in market interest rates. In these cases the initial interest rate is called a teaser rate.
Graduated Payment Mortgages
GPMs call for escalating mortgage payments. Payments start low and end high. Most real world GPMs call for a period of constant lower mortgage payments, perhaps three to five years and then an adjustment to constant higher payments for the remainder of the life of the loan. Under a GPM with low payments for 5 years and higher payments for the remainder of the loan, payments would have to satisfy the following expression where PAY is replace by PL and PH. This equation involves five unknowns, M, R, N, PL, and PH.
To solve for initial payments in terms of M, R and N, one needs to specify the relation between PL and PH. For example, if mortgage payments will increase by 20% after five years, then PH = 1.2 PL.
A simple form of GPMs for analytic manipulation is one that calls for mortgage payments to grow at rate g each period, that is PAYt = P(1+g)t. In this case our basic formula can be written as
and the initial mortgage payment, PAY1 = P (1+g) , can be written as
If g = 0 this expression reverts to the earlier equation for level payment mortgages. Figures 4 and 5 show the outstanding loan balance and the division of mortgage payments as between interest and principal for a GPM where R = 9% and g = 3%. Note that the loan balance increases initially. The hump pattern reflects the fact that for this example initial loan payments are not sufficient to cover interest due. The initial unpaid interest gets added to the loan balance.
GPMs were developed as a possible alternative during periods of high inflation. The inflation premium built into the mortgage rate is in anticipation of future increases in prices. Under fixed payment mortgages, the inflation premium increases mortgage payments from the very beginning of the loan before much inflation has occurred. GPMs postpone increases in mortgage payments until later in the life of the loan when it was expected that inflation will have increased wages and salaries making the increase in mortgage payments easier to handle. If the graduation factor, g, matches the rate of inflation and this inflation applies equally to income and house prices, then income will increase in step with mortgage payments and increases in house prices will more than match the increase in the outstanding loan balance during the early years of the loan. If inflation is less than the rate of graduation or if it does not apply equally to income and house prices, homeowners could find themselves in a difficult position.
Outstanding Balance: Level Payment Mortgage
M = $100,000; N = 25; R = various
Figure 2
Mortgage Payments: Level Payment Mortgage
M = $100,000; R = .09; N = 25
Figure 3
Mortgage Payments: Level Payment Mortgage
M = $100,000; R = .15; N = 25
Outstanding Balance: Graduated Payment Mortgage
M = $100,000; R = .09; g = .03; N = 25
Mortgage Payments: Graduated Payment Mortgage
M = $100,000; R = .09; g = .03; N = 25
Figure 6
Tradeoff Between Points and Contract Mortgage Rate
Contract Mortgage Rate