Having trouble understanding debt repayment maximization

[Chase S.]

I consider myself a financially savvy person, but I came across a debt repayment question that literally every person disagrees with me on (online and in person).  So I am trying to figure out why I am wrong or why everyone else thinks they are right.

It essentially boils down to how to handle almost paid off mortgages in terms of debt repayment, namely use the loan term rate or create a new effective rate that you calculate.

For example, if you have two mortgages, a 50k at 5% with 30 years left and a 10k at 7% with 2 years left and you have some windfall (nominally 5k) to contribute to one balance, which should you apply it to to maximize the effect.

Every person (including the snowball and avalanche methods) say pay off the 10k at 7% since it is the lower balance with a higher interest rate.  Since these are mortages, why doesn't applying it to the significantly longer balance make sense due to the amortization of the loans?  I am arguing from a mathematical optimization perspective and not the psychological boost of the snowball.

Since it is amortized, the interest is front loaded and the last few years should be primarily principle only.  Shouldn't that turn the 7% loan into effectively a <2% loan (and potentially <0.5% for the last year) since you are paying minimal remaining interest thus any applied windfall would not save you money in future interest payment?  Why is it treated as a uniform 7% through the life of the loan for all debt payoff conversations instead of different effective interest rate 'brackets' depending on the remaining time of the loan? IE, if you have a 30 year mortage and intend to sell in 5 years, you have paid more than 6% effective interest upon sale whereas in the last year, you have paid essentially 100% principle and virtually 0% interest.

I hope the question makes sense, if not, let me know.

Thanks
 

[thd7t]

Ignoring the topic of mortgage pay down here, the reason to pay the higher interest debt is a little more accessible by looking at it in a modular sense:

1x10k @7%
5x10k @5%

Even though you pay a greater amount of principal at the end of a loan, you're still paying the same rate of interest on the balance.

[Chase S.]

The topic is specifically mortgage pay (potentially combined with other loans, but just sticking with multiple mortgages for simplicity).

The entire loan was averaged out to reach the term rate, thus the rate near the end of life is effectively lower since most of the interest was already paid in the beginning.

A generic Amort table (100k, 5%, 30 years) shows:
Year 1: ~1500 in principle, ~5000 in interest
Year 28: ~5600 principle, ~750 interest
Year 29: ~6000 principle, ~500 interest
Year 30: ~6300 principle, ~200 interest

paying the $5k at year 29 would save maybe $500 tops in interest whereas the other payment would save thousands over 30 years.  One calculator I used showed it would save about 17k and end the loan a few years earlier.

 

[Jrr85]

The topic is specifically mortgage pay (potentially combined with other loans, but just sticking with multiple mortgages for simplicity).

The entire loan was averaged out to reach the term rate, thus the rate near the end of life is effectively lower since most of the interest was already paid in the beginning.

A generic Amort table (100k, 5%, 30 years) shows:
Year 1: ~1500 in principle, ~5000 in interest
Year 28: ~5600 principle, ~750 interest
Year 29: ~6000 principle, ~500 interest
Year 30: ~6300 principle, ~200 interest

paying the $5k at year 29 would save maybe $500 tops in interest whereas the other payment would save thousands over 30 years.  One calculator I used showed it would save about 17k and end the loan a few years earlier.

You are assuming that you drop your payment amount when you payoff the $10k.  If you assume that, then yes, you could pay less interest by applying the $5k to the larger loan, because it effectively keeps you paying the larger monthly payment (i.e., the combined amount of the payments on the $50k and $10k loan), so that's enough to offset the fact that you apply the $5k in a non-optimal way. 

Most people when they look at that question, they just assume you will keep paying the same monthly amount.  So paying off the $10k earlier, and then taking the payment on the $10k and starting to apply it to the larger mortgage when you pay off the $10k results in the maximum interest savings because they payoff the higher interest rate debt first. 

ETA:  Looking at it another way.  You can use the $5k to minimize interest payments, or to maximize cashflow.  If you want to minimize interest payments, you pay it towards the $10k loan, because that saves you the most interest for the $5k.  If you want to instead focus on maximizing cashflow, you still apply it to the $10k because that allows you to pay it off faster and frees up that monthly payment to go to other things. 

Paying the $5k to the $50k loan is the worst of both worlds.  It doesn't free you of the payment on the $10k faster, and it doesn't save interest in an optimal way.  The only thing that can be said for it is if you are concerned you are not disciplined enough to pay more on the $50k loan when the $10k loan is paid off, it forces you to pay a bigger amount for longer, effectively reducing your long term interest costs if you assume you will be undisciplined.

[Capt j-rod]

I always consider multiple things when it comes to eliminating debt... There is some satisfaction in paying something off and being freed of the responsibility of that note. Cash flow is another thing that matters. Having a better monthly cashflow allows for better opportunities when they are available. I have paid off a 0.9% car loan just to get the $$$ free and be able to direct it to where it served me the best. During this time I was carrying a mortgage of 3.5% fixed. Long term I should have reduced the mortgage, but short term I now have $300 more to pay it off. If all of your financed are dialed in and you have no "really stupid debt" aka charge cards then you are really splitting nickels at this point. I have my own business so cash flow means a lot to me. I guess either is fine, but I would kill the small loan then attack the big one with the flexibility of having some extra monthly emergency $$$ if life throws you a curve ball.

[secondcor521]

I consider myself a financially savvy person, but I came across a debt repayment question that literally every person disagrees with me on (online and in person).  So I am trying to figure out why I am wrong or why everyone else thinks they are right.

It essentially boils down to how to handle almost paid off mortgages in terms of debt repayment, namely use the loan term rate or create a new effective rate that you calculate.

For example, if you have two mortgages, a 50k at 5% with 30 years left and a 10k at 7% with 2 years left and you have some windfall (nominally 5k) to contribute to one balance, which should you apply it to to maximize the effect.

Every person (including the snowball and avalanche methods) say pay off the 10k at 7% since it is the lower balance with a higher interest rate.  Since these are mortages, why doesn't applying it to the significantly longer balance make sense due to the amortization of the loans?  I am arguing from a mathematical optimization perspective and not the psychological boost of the snowball.

Since it is amortized, the interest is front loaded and the last few years should be primarily principle only.  Shouldn't that turn the 7% loan into effectively a <2% loan (and potentially <0.5% for the last year) since you are paying minimal remaining interest thus any applied windfall would not save you money in future interest payment?  Why is it treated as a uniform 7% through the life of the loan for all debt payoff conversations instead of different effective interest rate 'brackets' depending on the remaining time of the loan? IE, if you have a 30 year mortage and intend to sell in 5 years, you have paid more than 6% effective interest upon sale whereas in the last year, you have paid essentially 100% principle and virtually 0% interest.

I hope the question makes sense, if not, let me know.

Thanks

If, by "maximiz[ing] the effect" you mean minimizing overall interest paid, then everyone else is right and you are wrong.  Let me try to explain why.  Mostly I think it's because you misunderstand how amortization works.  What follows assumes a typical US first and second mortgage; it may work differently in unusual cases in the US or in other countries.  I'm also ignoring escrow accounts, since those are immaterial to the discussion.

In a typical simple case where one is just paying a monthly mortgage on a regular basis, the interest calculation is simple.  They take the current balance on the mortgage, multiply it by the interest rate, divide by 12 to get a monthly figure, and that's your interest due that month.  They take your payment, subtract the interest due, and whatever's left gets applied to principal.  This process gets repeated every month.

What this means is that you're borrowing at 7%, all the time, every month, every day, every dollar, no matter what the term of the loan is or the outstanding balance.  There are no "effective brackets" or <2% loans or any of that.

The reason that you pay less interest at the end of the mortgage compared to the beginning is simply due to the fact that at the end of the mortgage you owe the bank less.  There is no magic going on with 7% turning into other interest rates.  Consider this simple example:  you have a $100K mortgage at 7% interest and I have a $1M mortgage at 7% interest.  I pay more interest than you every month, but that's because I owe the bank more money than you do, not because of any fancy amortization stuff or term of the loan or anything else.

In the case of your two mortgage example, for every $1K you have on your 5% loan, your monthly interest amount is $4.16 ($1000 x 5% / 12).  For every $1K you have on your 7% loan, your monthly interest amount is $5.83 ($1000 x 7% / 12).  This is true irrespective of the amounts of each loan and irrespective of the original or remaining term on each loan.

So obviously, to minimize interest, you would want to take each $1K chunk and apply it to the costlier loan - you'd rather save $5.83 in interest over $4.16 in interest.

[Chase S.]

You are assuming that you drop your payment amount when you payoff the $10k.  If you assume that, then yes, you could pay less interest by applying the $5k to the larger loan, because it effectively keeps you paying the larger monthly payment (i.e., the combined amount of the payments on the $50k and $10k loan), so that's enough to offset the fact that you apply the $5k in a non-optimal way. 

Paying the $5k to the $50k loan is the worst of both worlds.  It doesn't free you of the payment on the $10k faster, and it doesn't save interest in an optimal way.  The only thing that can be said for it is if you are concerned you are not disciplined enough to pay more on the $50k loan when the $10k loan is paid off, it forces you to pay a bigger amount for longer, effectively reducing your long term interest costs if you assume you will be undisciplined.

This is the concept I am trying to overcome looking at the math when talking to others. I honestly am not trying to create an argument, but honestly hash out the real answer.  For even simpler comparison purposes (trying to prove the concept over specific situation) lets reduce it to 100k 5% 30 year mortgage (A) and 100k 7% 30 year mortgage with 2 years left (B) [thus current balance ~15k]. I have compared a handful of amortization tables and Payoff calculators for the math.  The two scenarios are apply 5k at the beginning of (A) and adding the (B) payments in at year 28 OR applying 5k to (B) for a 10 month earlier payoff and adding those payments to (A) at year 28, 10 months.  Monthly payment for (A) is $537 and (B) is $665.

The total principle payoff is the same.  The only differences are the time until all loans are paid off and the total future combined interest payments.
If I pay 5k to (A) at the beginning of 30 years, the total interest paid is just under $30k. If I pay the 5k to (B), the total interest paid is over $39k.  The payoff in both cases is basically 8 years (they are 1 month apart).

This is the math that tells me paying (A) is the smarter decision despite literally everyone stating pay (B). Both loans get paid off in 8 years with the difference being 9k in interest if you pay the mortgage near the end of it's life.  And that is the reason I am asking this question.

[Chase S.]

I will also concede that I am a pay the higher interest person over the snowball method. This question could also come from misusing the calculators or misapplying something. 

I am mainly just trying to hash out a math anomoly I noticed after using some online calculators to answer another question I saw.

If the calculators were right (and I used them correctly), then the avalanche concept isn't the most efficient. If I used the calculators (maybe 3 different sites) incorrectly, then the avalanche would still be the most efficient.

[MDM]

...lets reduce it to 100k 5% 30 year mortgage (A) and 100k 7% 30 year mortgage with 2 years left (B) [thus current balance ~15k]. I have compared a handful of amortization tables and Payoff calculators for the math.  The two scenarios are apply 5k at the beginning of (A) and adding the (B) payments in at year 28 OR applying 5k to (B) for a 10 month earlier payoff and adding those payments to (A) at year 28, 10 months.  Monthly payment for (A) is $537 and (B) is $665.

The total principle payoff is the same.  The only differences are the time until all loans are paid off and the total future combined interest payments.
If I pay 5k to (A) at the beginning of 30 years, the total interest paid is just under $30k. If I pay the 5k to (B), the total interest paid is over $39k.  The payoff in both cases is basically 8 years (they are 1 month apart).

With no extra payments,
               Total Interest
Loan A         93,256.50
Loan B       139,510.98

With a $5K extra payment in the first month
               Total Interest
Loan A         77,692.39
Loan B       109,864.81

You save $15,564.11 by putting the $5K to loan A, or $29,646.17 by putting the $5K to loan B.

Is that what you see also?

I didn't follow the "Both loans get paid off in 8 years" part.

[Jrr85]

You are assuming that you drop your payment amount when you payoff the $10k.  If you assume that, then yes, you could pay less interest by applying the $5k to the larger loan, because it effectively keeps you paying the larger monthly payment (i.e., the combined amount of the payments on the $50k and $10k loan), so that's enough to offset the fact that you apply the $5k in a non-optimal way. 

Paying the $5k to the $50k loan is the worst of both worlds.  It doesn't free you of the payment on the $10k faster, and it doesn't save interest in an optimal way.  The only thing that can be said for it is if you are concerned you are not disciplined enough to pay more on the $50k loan when the $10k loan is paid off, it forces you to pay a bigger amount for longer, effectively reducing your long term interest costs if you assume you will be undisciplined.

This is the concept I am trying to overcome looking at the math when talking to others. I honestly am not trying to create an argument, but honestly hash out the real answer.  For even simpler comparison purposes (trying to prove the concept over specific situation) lets reduce it to 100k 5% 30 year mortgage (A) and 100k 7% 30 year mortgage with 2 years left (B) [thus current balance ~15k]. I have compared a handful of amortization tables and Payoff calculators for the math.  The two scenarios are apply 5k at the beginning of (A) and adding the (B) payments in at year 28 OR applying 5k to (B) for a 10 month earlier payoff and adding those payments to (A) at year 28, 10 months.  Monthly payment for (A) is $537 and (B) is $665.

The total principle payoff is the same.  The only differences are the time until all loans are paid off and the total future combined interest payments.
If I pay 5k to (A) at the beginning of 30 years, the total interest paid is just under $30k. If I pay the 5k to (B), the total interest paid is over $39k.  The payoff in both cases is basically 8 years (they are 1 month apart).

This is the math that tells me paying (A) is the smarter decision despite literally everyone stating pay (B). Both loans get paid off in 8 years with the difference being 9k in interest if you pay the mortgage near the end of it's life.  And that is the reason I am asking this question.

Just to show you the math with your example.  2 loans:

Loan 1: 100k, 5% interest, 30 year term, $536.82
Loan 2: $14,859.54, 7% interest, 2 year term, pmt of $665.30 (this is the last two years of the 100k, 7% loan for 30 years). 

Apply the $5k to Loan 1, effectively turning it into a $95k loan.  Continuing to make the same 536.82, and after two years, your 7% loan will be paid off, allowing that $665.30 to be applied to Loan 1.  Then you pay it off in another 7.64 years for a total of 9.64 years of payment, resulting in total interest payments on the $100k loan of $28,098, plus $1,107.66 for the $14,859.54 loan for a total of $29,205.66 in interest paid. 

Conversely, if you apply the $5k to the $14,859.54 loan, you pay that off in 1.3 years, and payh $482.52 total interest.  If you then apply the $665.30 to the $100k laon, you end up paying it off in another 8.32 years for a total of 9.6 years of payments, and pay a total of $29,084.75.

Basically tiny savings of $120.91 in interest.  Which makes since because you were only going to carry that 7% loan for at most two years, and two years of 2% interest savings would be $200 (and obviously the final result would be less than that since you wouldn't be carrying that $5k balance for the two years, so it'd be just over half that savings because of the accelerated paydown of principle in the second year).