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====== Math 445 Lab 4: Loan Calculator ======

Time Estimate: 2 hours

Important MATLAB commands that will be used: ''fzero''.


Buying a car can be tricky and some salesmen are just plain
dishonest. My brother went to go buy a car some time ago in a diﬀerent
state while he was still a student at a university.  The salesman was
someone he knew and trusted. They looked at several cars and each time
the salesman would quote a cost for the car and a price per month on a
5 year loan.  

It just so happens that my brother was married at the
time and his wife has some unique ﬁnancial training. When she started
college, she got a job at the local university credit union.  This
credit union only had two employees, the president of credit union and
one assistant (my brother’s wife). After she had worked there for
about two years, the president of the credit union resigned and she
was made the president. She had been in that position for a little
over a year, handling every major transaction of the credit union,
when they went to look at buying a new car. She had a ﬁnancial
calculator with her and would calculate what the monthly payments
should be on the car based on the price and interest rate. They
quickly realized that the salesman was quoting them monthly payments
considerably higher than they should have been.  They confronted the
salesman, whom they had previously trusted, and asked him what was
up. 

At this point the salesman told them that it was a game that
they played. They would tell the customers a monthly payment that was
too high so that when they sat down with the manager who asked them to
buy an extended warranty they could say that it would only be a little
bit more per month. Basically they were lying about the original cost
per month to trick customers into believing that the extended
warranties were cheaper than they actually were.  

In this lab we are
going to write a MATLAB function to calculate monthly loan payments.
I highly suggest that whenever you go to make a purchase with a loan
or take out any loan that you make sure you know exactly what is going
on. Run this program that you have written or go to the website of
almost any bank and they should have a loan calculator. The above
story is true, and it is not my only story of people being deceived on
major loan purchases. My brother was lucky because his wife knew some
math.

--Prof. Mark Lyon
==== Problem 1 ====

We ﬁrst need to identify what the inputs to our function will be. We
will need to have the total number of payments in months or ''term'' of
the loan. We will need the interest rate, ''rate'', as well as the loan
amount, ''loan''. We will assume the interest is compounded monthly
(standard for loans today, savings account interest is typically
compounded daily).  Rather than have any outputs, we will just print
our results to the screen. Save the function as ''LoanCalc.m''.  Start
by beginning the function with the command function
''LoanCalc(loan,term,rate)'' Hit enter until you come to the line number
15 and put ''end'' there.

We are actually going to start with a simpler problem and let MATLAB
solve for the harder one for us. We are going to write a sub-function,
or a function within a function. Some where around line 8, start
another function (call it ''LoanPay'', with input of just ''payment'' and
a single output, ''balance''.) This function is going to compute, given
a loan with a certain rate, what the balance due on the loan is going
to be after all the payments of amount ''payment'' have been made. This
function will need an ''end'' before line 15. In the function, ﬁrst let
balance=loan; Then put a ''for'' loop over the ''term'' months. With each
month (or with each iteration of the loop) the balance should be
increased by the interest rate times the balance and reduced by the
amount of the payment. Test this sub-function by running it (on line
5) by the command ''bal=LoanPay(300)'' which should then give ''bal=-1151.44''
if you have set ''format bank'' when you enter ''LoanCalc(15000,60,0.05)''
in the MATLAB command window. Verify you have completed this step with
the TA.

==== Problem 2 ====

The returned balance in the previous step was negative because the
payment of 300 per month was too high. Changing the payment to 200 per
month would show a positive balance.  How about 250? What would you
try next? We could keep trying diﬀerent values until we zeroed in on
the correct value, or we could use a built-in MATLAB function that
does the same thing. It is called ''fzero'' and it ﬁnds a value that
will result in the function returning a zero.  On line 2 of your code
put the line ''payment=fzero(@LoanPay,[0,loan])'' and then change line 5
to read ''bal=LoanPay(payment)'' and run this with the command
''LoanCalc(15000,60,0.05)'' to get the results:

<code>
payment=283.07
bal=-0.00
</code>

Verify you have completed this step with the TA.

==== Problem 3 ==== 

Add some additional code to your ''LoanCalc'' function to calculate the total amount of
interest paid during the life of the loan. Use this code to answer the following questions:

**a.** If you borrow %%$%%15,000 for 5 years at a 5% interest rate, how much interest would you end
up paying?

**b.** If you borrow %%$15,000%% for 5 years at a 10% interest rate will you end up paying exactly
double in total interest compared to a 5% interest rate?

**c.** A typical home in the area around Durham, NH costs around %%$300,000%%. If you pay 
%%$50,000%% on the down payment and you borrow the remaining %%$250,000%% for 30 years at a 4% 
interest rate, how much interest would you end up paying? How does that amount compare 
to the original loan?

**d.** How much would your payments increase by (in amount and percent) if you decided to
pay off the mortgage in just 15 years instead of 30? How much would you save interest?

**e.** How much would you pay in interest if you borrowed %%$250,000%% at a rate of 4% for 30 years
compared with how much you would pay interest if you borrowed %%$250,000%% at a rate of
8% for 30 years. How do your payments compare for these two different scenarios?
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