Loan and Mortgage Analysis and Comparison.
Foreword
Some years ago I was watching one of those documentaries about life in remote places
of the world and they were talking to a goatherd tending to his goats somewhere remote
in Africa who was being helped by his very young son. The interviewer asked the
man why his son did not go to school to which the man replied that there was no point
in sending him to school because anything he learnt in school would be useless to their
life herding goats. I immediately thought that the man had it backwards because
if they had an education they would not be herding goats in the first place.
I hear so many people complain that what they teach in schools is useless and not worth
learning because it is no help in later "real life". These people later go on to
minimum wage jobs and to complain about "jobs being exported" and about "immigrants
taking our jobs and driving wages down".
I am not the first or the only one to be concerned about the sorry state of schools and
the education they are providing but I believe this is part of a wider phenomenon of
general lack of appreciation of education and knowledge in society.
People get loans and mortgages and find themselves totally unable to analyze and compare
and judge and this is just because they slept through math class in school. I am
always surprised by the innumeracy of people, even people who have a certain education,
not to mention the uneducated. Of course, it's easy for me to say as my background
is in engineering.
A bit of history
Banks used to give loans (mortgages) with a prepayment clause which subjected the borrower
to a penalty if the loan was paid off early. This made sense to the lender because
they had done all the upfront work of setting up the loan with the expectation that the
loan would be productive over the entire agreed period. But if it was paid early
the bank had only obtained part of the interest and now had the principal and had to do
all the work to lend it again. But the borrowers did not like this because when
the rates went down they would refinance and pay off the original loan and they hated
paying a penalty. Pressure was put on politicians and prepayment penalties were
outlawed in many states but lenders found an easy way to get around this in the form of
"points" and "loan origination fees" as we shall see.
Loan points and fees are a prepayment penalty
Let us analyze a simple example. Suppose you ask me (A) to lend you $100 and I agree
to lend you this amount at no interest but with a $5 "origination fee" so you give me
$5, I give you $100 and you agree to pay me 5 monthly installments of $20. No
interest? Wait! Let me give you a different scenario so we can compare.
You ask your friend Bob and he says he will lend you $95 at 20.82% interest which
works out that you have to pay him back 5 monthly installments of $20. So you see
in both cases you walk out with a net loan of $95 and in both cases you pay back 5
monthly installments of $20. So you can see that up to this point in the analysis
both loans are identical and the "origination fee" is, in fact, interest paid on the loan
of $95 except the lender is calling it something else and trying to get you to see it
as something else.
So which loan should you choose? Are both loans identical?
mo. 
A 
B 
Diff. 
0 
100.00 
95.00 
5.00 
1 
80.00 
76.65 
3.35 
2 
60.00 
57.98 
2.02 
3 
40.00 
38.98 
1.02 
4 
20.00 
19.66 
0.34 
5 
0.00 
0.00 
0.00 

If both loans are paid over the full life of 5 months then yes, they are identical
but suppose after two months you have some extra money and would like to pay back
the full amount you owe at that point. Look at the table on the right.
At the end of month 2 with loan A you owe $60 while with loan B you owe $57.98 so
if there is any chance of prepayment you are better off with loan B. To make
loan B equal to loan A we would establish a prepayment penalty for loan B which would
be the difference between the principal owed at each month of loans A and B (i.e.
the column labeled "diff."").
In summary, a loan of $100, with 0% interest rate, $5 origination fee, payable in
5 monthly installments of $20 is exactly the same in every regard as a $95 loan,
with 20.82% interest rate and a prepayment penalty which starts at $5 and gradually
diminishes according to the table until it is $0 at the end of the five months.
Rather than use a complicated formula the lender would have used a linear decreasing
function, in this case $1 per month of prepayment. That works even better for
the lender and is simpler to calculate.
Since prepayment penalties are not allowed but the bank charges points and fees,
in order to make a meaningful comparison between loans we need to reduce them
to the same parameters so that we can compare interest rates and prepayment
penalties in a meaningful way.
Formula
Any spreadsheet or financial calculator can easily calculate the monthly payment
of a loan given the relevant parameters but I post the formula here so you don't
have to go looking for it all over the Internet. It is pretty much the only
formula you need to analyze a loan.
K is the amount of the loan (das kapital).
i is the interest rate per period. We assume the period
is monthly and as the interest rate given is annual we divide it by 12 so, for
instance, if the interest rate is 6% p.a. then
i = 0.005. n is the
number of periods in the loan. In the case of a 30 year loan it would be 360.
Using this formula we can calculate that a loan for $105128.20, 3% interest rate,
360 months will have a monthly payment of $443.22 and a loan for $100000 at 3.50%
and also 360 months will have a monthly payment of $449.04. We will use these
examples in the following section.
While you can use the formula above Microsoft Excel spreadsheet has a builtin
function, PMT, which calculates the payment.
Simplified Syntax: =PMT(rate,nper,pv) example: =PMT(0.05/12,12*15,100000) = 790.79
Another useful concept is that of
ceteris paribus
which should be familiar to students of economics, science and engineering.
To compare two loans we will adjust to equalize the net amount of the loan and the payment
period and then we can meaningfully compare the dollar amounts paid over time.
Lender 
Points 
Fee 
Interest rate 
Mo. payment 
A 
2.5 
$2500 
3.00% 
$443.22 
B 
0 
$0 
3.50% 
$449.04 

A real world example
Two lenders, A and B, offer 30year loans with the terms in the table at right.
(The monthly payments are calculated for a net loan of $100,000 in both cases.)
How do these terms compare? Which one is more advantageous for the borrower?
Try asking this question of even well educated people and almost nobody will be able
to give you a really good analysis. Even loan officers will usually just mention
rules of thumb which really are mostly misleading. I have seen this when people
are considering refinancing. The fact is that most people just do not have a clue
when it comes to analyzing these things.
So which one of the two loans is best? The answer is "it depends" as we shall
soon see.
In the case of lender A the nominal amount of the loan would be $105,128.00 which after
deducting points and fees results in a net loan of $100,000. Applying the payment
formula noted earlier a loan of $105,128.00 at 3% p.a. payable in 360 monthly installments
would require a monthly payment of $443.22. The actual terms of the loan are $100,000.00
principal at 3.40% interest rate which over 360 months results in the same monthly payment
of $443.22.
In the case of lender B there are no points or fees so the net amount of the loan equals
the nominal amount ay $100,000.00. With an interest rate of 3.5% p.a. and the same
360 months the monthly payment will be $449.04.
So, a simple, first order comparison shows that with loan A we save $5.82 monthly but we
pay $5000.00 upfront in points and fees. At first blush it would seem it is not
worth paying $5000.00 upfront in order to save $5.82 monthly but this is misleading.
Let us look at the two extreme cases. If you sign the loan and immediately change
your mind and prepay it back then loan A is clearly worse because you have to pay back
$105,128.20 whereas with loan B you only have to pay back $100,000.00. In this case the
nominal interest rate of the loan is irrelevant and the points and fees are, as we saw,
in efect, a prepayment penalty.
If we keep the loan for the full 360 months then loan A wins because of the lower monthly
payment. Initially loan B is better but after 360 months loan A is better.
There is a point in time before which prepaying means loan B saves money but prepaying
after that point means loan A saves money. So the next question is how do we find
out that point in time and what is the probability that we will prepay the loan before
that point.
As a different analysis instead of equalizing the payments at 360 months we could equalize
the amount of the monthly payment to the higher amount of $449.04 and see how many monthly
payments would be required to amortize the loan. In the case of loan A if we paid
$449.04 monthly we would amortize the loan in 352.4 months so we would save 7.5 months of
payments with respect to loan B. In this particular example the difference between
loans A and B are not huge but in real world comparisons they can be very significant.
