math modeling and R code
MesaLs4
Sample2-Project.pdf
Chrysalis Science Consulting �
How much can I afford?
Choosing What’s Right for You
Presented by: Chrysalis Science Consulting
For use by: SSquared Bank & Co.
(SS Banks)
Mortgage Resources
Chrysalis Science Consulting Allison Langley ChrysalisScience.com 5500 Campanile Drive San Diego, CA 92182-0828 [email protected] (858)414-8041
March 3, 2017
Chrysalis Science Consulting �
Executive Summary
Objectives This report showcases examples of resources to aid home-buyers in
understanding the mortgage process and the large effect of a slight change
on their cash flow. This clear display of information will ease many of the
concerns that home buyers share when searching for a loan.
Goals The duration of this report will focus on the benefits SS Banks partnering
with Chrysalis Science Consulting. The advantages that SS Banks will
gain on the home-mortgage market are illustrated to the reader with
examples of resources that SS Banks will have the ability to use and, as a
result, so greatly increase their customer retention rate.
Conclusions Partnering with Chrysalis Science Consulting will significantly raise
customer satisfaction with SS Banks’ transparency. This report will show
how potential customers will find confidence in knowing how the mortgage
works and will easily understand how SS Banks’ low interest rates yield
such a huge advantage over choosing competing banks.
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Chrysalis Science Consulting �
Introduction
Problem Identification and Modeling the Mortgage Chrysalis Science Consulting will work with SS Banks to secure the highest
possible number of potential borrowers. This will be primarily achieved with
easily understood displays of information, charts, and breakdowns of the
total cost associated with mortgages. Many borrowers don’t realize the
differences and advantages of various mortgages. This report will give
examples of tools that may be offered when partnering with Chrysalis
Science Consulting to ease the mortgage process and aid customer
understanding of how their cash flow is affected depending on the type of
loan chosen. We will assume a principal of $200,000 and begin analysis of
a 30-year term loan at an annual interest rate of 3.375%. This will then be
compared and contrasted with a 15-year term at the identical rate. We will
conclude with sensitivity analysis and include a 20-year term to show the
resulting affect on total interest paid and overall cashflow, as well as the
huge effect that a slight change in interest rate poses. These resources will
all hep to ensure that more individuals choose SS Banks over competing
banks.
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Mortgage Abstraction In order to begin analysis on a 30-Year mortgage we must first define the
variables that will be in use:
Principal = (P)
• The total amount of money borrowed from the bank
Total number of payments you will make = (n)
• The loan will be paid off in monthly payments
• We are first considering a 30-year mortgage, meaning that after a 30
year period, the loan and interest will be completely paid off
• In this case, 12 payments each year, for 30 years: n = 360 months
Your monthly payment = (x)
• The amount of money you will spend each month until the loan is
paid off
Principal remaining at the end of the kth month = (Pk)
• At the end of the 30 month period (a total of 360 months), k=360
• Because the loan will be paid off after 360 months, when k=360, Pk
(the principal remaining) will equal zero
Monthly interest rate = (r)
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• Mortgage is paid every month, meaning that interest is also paid
every month
• The interest rate of a mortgage is the annual rate
• To calculate the monthly interest rate, the mortgage rate is simply
divided by 12
r = 3.375% ÷ 12 months
• r = 0.28125% = 0.0028125
Monthly payoff of principal = (Qk)
• Each month’s payment is divided between paying off the principal
and paying interest on the remaining principal
• At the beginning of the term, the percentage of the monthly fee that
pays off the principal will be smaller than the percentage delegated to
the interest
• As the loan matures, the percentage paying off the principal will
grow, and the interest will begin to decrease, until the loan has been
completely paid off
Monthly payment of interest = (Ik)
• This is the portion of a given month’s payment that is collected as
interest on the remaining principal
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• When the term begins, a higher portion of the monthly payment will
go towards inters, but as the loan matures and more payments are
made, this percentage will decrease and the majority of the
payments will begin to go towards the principal
The establishment of these variables allows us to now model the mortgage
with a set of equations. Creating equations that represent the life of the
mortgage will help us to clearly answer questions that customers may have
regarding their monthly payments, principal, interest, and overall cashflow.
Data and Method
Equations of the Mortgage Model Calculation of the Monthly Payment
• The monthly payment (x) is determined by the original loan amount
(P), monthly interest rate (r), and the total number of payments (n)
• Each monthly payment will be identical to this amount, every month,
until the entire loan is paid off
Principal Remaining Each Month
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• Using this equation, we may calculate the remaining principal after
the kth month
Payment Towards Principal and Interest
• This equation clearly illustrates that the monthly payment is
composed of both an interest payment (Ik) and a payment towards
the remaining principal (Qk).
Solutions of the Model Our first model to share with the customer will be of a fixed rate, 30-year
term loan of $200,000, at SS Banks’ annual interest rate of 3.375%.
The Variables:
Principal (P) = $200,000
Number of months (n) = 360
Monthly interest rate = 0.0028125
Referring to the equations previously defined, customers are able to
confidently work in parallel with SS Banks, to calculate different aspects of
their mortgage. Customers can now easily calculate a monthly payment of
$884.19. Each month, this number will remain the same. A portion of your
payment goes towards paying off the principal, and a portion goes towards
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+Q I
Chrysalis Science Consulting �
paying the bank interest on the loan. It is this allocation that begins to
change over time. The very first payment has interest on a total of
$200,000 dollars, meaning that $562.50 goes towards interest and $321.69
of the principal is actually paid off. Using the equations above, we can see
that, the next month, $199,678.31 is now owed to the bank. Interest is only
paid on the remaining principal. So as this amount goes down, the interest
also goes down, and a larger percentage of the $884.19 monthly payment
begins to go towards the principal. The graph below displays the total
interest paid over the life of a 30-year loan.
� Page � of �8 1430−yr
30−Year Term Loan
A m
ou nt
($ )
0 50
00 0
10 00
00 15
00 00
20 00
00 25
00 00
30 00
00 Principal Borrowed
Interest Paid
Total Dollar Amount Spent
Chrysalis Science Consulting �
Alterations in Term Length Remaining with the same type of mortgage (fixed-rate,
at 3.375%), we can now alter the term length and
observe the resulting effect on interest and overall cashflow. Utilizing the
same equations, we calculate the monthly payment of a 15-year term to be
$1,417.52. While this is significantly higher than the monthly patent of the
30-year term loan, the amount of interest is also greatly reduced. The
figure below easily displays this difference in total interest paid when
considering a 15-year loan over a 30-year loan.
� Page � of �9 1430−yr 15−yr
Interest on 30−Year and 15−Year Terms
A m
ou nt
($ )
0 50
00 0
10 00
00 15
00 00
20 00
00
Principal Borrowed
Total Interest Paid
Chrysalis Science Consulting �
If the buyer can afford the higher monthly payment, then the shorter 15-
year term may be preferred. Because the loan is paid off in half of the time,
the interest does not grow to be as large as the previously considered loan.
However, if the borrower will be challenged to make such high monthly
payments, the 30-year term may be a better option. The 30-year term
increases the monthly cashflow of the borrower. This means that if the
customer is in need of the extra money each month, the 30-year loan will
pose a relatively small burden. If the borrower is capable of paying the
higher payments of a 15-year loan, they may still prefer to choose a 30-
year term to allow for further
investment. The additional
cashflow may be used to invest in
other opportunities, and make
additional money before being
spent on the loan. Choosing a
20-year loan may be an additional
option that the home-buyer would
like to consider. To the right is a
graph depicting differences
between these three options.
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30−yr 20−Yr 15−yr
Interest on 30, 20, and 15−Year Terms
A m
ou nt
($ )
0 50
00 0
10 00
00 15
00 00
20 00
00
Principal Borrowed
Total Interest Paid
Chrysalis Science Consulting �
The lower monthly rate of $1,147.11 may be more achievable than the rate
associated with a 15-year loan. The 20-year term cuts the total interest but
still allows for additional cashflow each month, allowing more the customer
to have more financial flexibility than under the 15-year loan, and a lower
interest payment than when paying for a 30-year loan.
Slight Changes in Interest Rate; Huge Changes in Payment
Now that the borrower more fully understands the differences associated
with options in term length, it is important to illustrate how the interest rate
so greatly affects their total payment. It may be hard for borrowers to
understand the large impact that an interest rate has because the changes
to the rate are so slight (less than 1%). By looking at the total interested
paid, we may much more easily see this large impact.
Interest Rate Monthly Payment Total Interest Paid 3.375% $884.19 $118,309.27
3.5% $898.09 $123,312.18 3.625% $912.10 $128,356.94 4.125% $969.30 $148,947.81
4.5% $1,013.37 $164,813.42
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Conclusion These resources will help to raise clients’ confidence and understanding of
mortgages and how to choose the packages that best fits their financial
needs. Additional information, such as the affect of payments in addition to
the monthly payment may also be included. SS Banks may want to have
additional equations and further derivations such as this:
One caveat this method is that some potential customers may shy away
from the mathematical equations. For those less inclined to mathematics,
reliance on visuals such as graphs and charts may be more welcoming. It
is optimal to have additional visuals available, as well as detailed equation
sheets, so that the bank representative may alter their pitch based on the
needs of the customer. Details on other aspects of the loan, such as risk
analysis, may further entice customers to continue forward with SS Banks
for their mortgage. Chrysalis Science Consulting will work with you to
create all of the needed materials to successfully retain the potential
customers who visit your offices regarding a mortgage loan.
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References
"Chase Bank - Credit Card, Mortgage, Auto, Banking Services." JPMorgan
Chase and Co. N.p., 2017. Web. 01 Mar. 2017. <https://
www.chase.com/>.
Shen, Samuel S.P. INTRODUCTION TOMODERNMATHEMATICAL
MODELING WITH R. N.p.: Wiley-Interscience, 2017. Print.
Appendix Sample Code for RStudio
####General Mortgage Calculator####
annualRate<-0.03375
#Principal
p<-200000
nYrs<-30
#Months
n<-nYrs*(12)
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#rate
r<-annualRate/(12)
#monthly payment
x<-((p*((1+r)^n)*r)/(((1+r)^n)-1))
x
####Interest Payment Month for 1####
interest1<-p*r
interest1
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