Loan and Mortgage
Analysis and Comparison.

    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 formula for amount of loan payment as a 
function of interest rate
    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 built-in 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
2.5  $2500  3.00%  $443.22 
$0  3.50%  $449.04 


A real world example
    Two lenders, A and B, offer 30-year 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.