APPM 2360 Lab #1: Mortgage 101

Due: October 5, 2015

1 Introduction

A pair of close friends are currently on the market to buy a house in Boulder. Both have obtained

engineering degrees from CU and have an understanding of diﬀerential equations but lack the

skills necessary to decide on what type of mortgage to take out on their new house. In the

following sections, you’ll ﬁnd some research they have done on commonly used mortgage

structures. However, they have become busy and cannot continue their analysis. Since you are in

Diﬀerential Equations this semester, they have decided to enlist your help in writing a report to

understand their options.

Your task is to read the following sections and complete the tasks found in Section 3. Your

friends were nice enough to list various problems they would like solved. However, you should be

careful to write your responses in a cohesive report.

2 Background Information

In this section you will ﬁnd information on some of the most prevalent mortgage structures.

2.1 Interest Compounding

Often, the interest of a loan is expressed as the annual rate, that is, the percent of the

outstanding balance that is charged as interest over a year. However, the frequency with which

the rate is applied to the current balance may vary. This frequency is how often the loan

compounds. If the interest is compounded annually, the formula to calculate the amount of money

owed after the ﬁrst year is

y(1) = (1 + r)y(0),

where y(t) is the outstanding balance after t years, and r is the annual interest rate.

How would this change if, instead, the loan compounded semiannually? Then, half the interest

rate would be applied to the loan value every 6 months.

y(.5) =



1 +



y(0)

y(1) =



1 +



y(.5) =



1 +



1 +



y(0) =



1 +



y(0)

This pattern continues for any frequency of compounding. That is, if a loan is compounded n

times per year, the value of the loan after 1 year is

y(1) =



1 +



y(0)

The more frequently that a loan compounds, the higher the value at the end of the year. However,

there is a limit as n goes to inﬁnity. The limit, which models a continuously compounding loan, is:

y(t) = y(0)e

This model can also be expressed as a diﬀerential equation:

(t) = ry(t), y(0) = y

If the borrower makes a monthly payment of p dollars, the model becomes:

(t, y) = ry(t) − 12p, y(0) = y

(1)

2.2 Adjustable vs. ﬁxed rate mortgages

Some mortgages use a ﬁxed annual rate, the r in the model. The interest rate which a borrower

receives depends on his/her credit score, the value of the house, the borrower’s income, the

duration of the loan, and several other factors. Fixed rate loans are sold with a minimum

monthly payment, which ensures that the loan will be paid oﬀ within the duration. The bank will

get all of the money back sooner with a shorter duration, so the rate is lower for this type of loan.

Adjustable rate mortgages start at a lower rate than a ﬁxed rate loan, but after a certain period

of time, the interest rate increases and becomes tied to one of several public indexes. This results

in the interest rate increasing above the rate that a ﬁxed-rate mortgage would have for the second

portion of the loan. Common ﬁxed rate periods are 3,5,7, or 10 years. Mathematically, this

results in r being a function of time r(t) instead of a constant.

3 Questions

Your friends recently found a house on the market and needed to borrow $500,000 from the

bank in order to purchase. Your friends are very careful when reading mathematics so in your

analysis you should include as much relevant work as necessary so your friends can follow. They

provided you a list of possible questions that they need answered.

3.1 Analysis of Fixed Rate Mortgages

Your friends have various options when choosing a mortgage structure. The following points

should help you know what to include in your analysis.

1. First, examine the eﬀect of continuous compounding on the value of a loan. Assuming that

r = .05 and the original balance is $500, 000, compute the total cost of the loan after 5 years

for loans compounded 1, 2, 4, 12 times per year, without any payments. Compare these

values to the cost of a loan compounded continuously.

2. Next, gain a broad understanding of the behavior of the loan value by determining whether

there any equilibrium solutions to (1). If so, what are they, and what is their stability?

What do these equilibria represent in real-word terms?

3. Determine the exact behavior of the loan in your friends’ situation by solving (1) using

separation of variables, with y(0) = 500, 000, r = .05.

4. The size of the monthly payment p that your friends are willing to make plays a large role in

deciding the type of loan they should choose. Use the solution to (1) to ﬁnd the correct p to

pay oﬀ a 10 year ﬁxed rate mortgage with rate 3% and initial debt of 500, 000. Do the same

for a 30 year ﬁxed-rate mortgage with rate 5%.

5. While having a low monthly payment is nice, you should warn your friends that there is

quite literally a price to pay for this convenience. We can determine the total paid by

summing each monthly payment over the duration of the loan. How much interest is paid in

the 30 year ﬁxed rate mortgage? The 10 year?

6. Buyers often choose to pay as much of the cost as they can up front so that they don’t have

to borrow quite so much. Might this option be worth it for your friends? How much money

would the borrower save in each case if he/she paid $50,000 down on the house? (That is,

the mortgage began at $450,000.)

7. What are the advantages and disadvantages of taking out a 30 year ﬁxed rate mortgage as

opposed to a 10 year mortgage?

3.2 Programming Euler’s method

Often times, the diﬀerential equations we wish to solve will be diﬃcult to solve by hand so we

enlist the help of a numerical scheme. Here we will utilize MATLAB to perform Euler’s method.

1. First we want to create each piece we need to calculate y

(t). Create functions r(t) and p(t)

which return a constant interest rate of 5%, and monthly payment 8, 000.

2. Now, we put the pieces together. Create a function y

(t, y), which takes in t and y as input

arguments, and outputs the value of y

at that point. Note that we can put the functions r

and p inside the function y

. This is known as a subfunction.

3. Finally, we must program Euler’s method. Use a step size of .01, and use a while loop to

run the method until the mortgage is paid oﬀ (y = 0). When will a mortgage initial for

$500,000 be paid oﬀ, with a constant interest rate of 5% and a monthly payment $8,000?

Include a plot of the solution y(t).

3.3 Analysis of Adjustable Rate Mortgages

Now we turn to the adjustable rate mortgages. Use the Euler’s method code for these problems.

You may also ﬁnd the ﬁnd command helpful. Type ‘help ﬁnd’ into the command window to learn

more. Suppose that a bank oﬀers an adjustable rate mortgage, with the rate ﬁxed at 3% for the

ﬁrst 3 years, and then the rate follows an index which behaves as r(t) = .05 + .002t + .003 cos(

t).

1. Suppose the borrower pays $3000 per month. How long will it take him/her to pay oﬀ the

mortgage?

2. What about if he/she pays $4500 per month?

3. How much interest is paid in each case?

4. Plot both scenarios on the same graph. Explain what is going on in it. How does the

interest rate aﬀect the graph?

3.4 Conclusion

1. What do you think accounts for the rise in popularity of adjustable rate mortgages over

ﬁxed rate mortgages? Which type of mortgage would you recommend to your friends who

are buying a house, and why?

4 Items to Remember

• Your friends want a complete report so you should write full sentences explaining the

questions posed and your responses. Don’t simply number your responses to individual

questions.

• All reports must be submitted to D2L by the due date in a .pdf format. Failure to do so will

result in a penalty. Any MATLAB code must also be submitted to D2L.

• MATLAB code is bulky and hard to read. If you would like to include it in your report, do

so in an appendix.

• Your friends are sticklers for the rules, so they really want you to follow all of the project

guidelines on the APPM 2360 webpage. Be sure to read these carefully before starting your

project!