Author Topic: Mr. Math and paying off your mortgage  (Read 47854 times)

LLL

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Mr. Math and paying off your mortgage
« on: April 13, 2017, 08:39:34 PM »
It distresses me that I have seen more than one person decide to not pay-off their mortgage early “Pay-off” because they think mathematically it is incorrect to do so. I have written this post to dispel the myth.

The typical argument is constructed as so, the average rate of return for the stock market (Nominal of 10.7% / Real return of 8.5%) is higher than the current interest rate of 4% on my 30-year mortgage. Therefore, I will have a higher net worth if I “Invest” extra money instead of making additional principle payments my mortgage. The problem is they consider risk a qualitative factor and dismiss it from the calculation.

When you include risk in the calculation, Pay-off or Invest are both shown to be reasonable and mathematically correct alternatives. That’s right, one is not a feel-good option that is mathematically incorrect. However, it’s really hard to measure, quantify, project, and display risk. But I decided to give it a try.

The Scenario:
Frank just bought a house with a 30-year, 4% interest rate, $165,000 mortgage. He has an extra $1,500 per month after maxing out all tax advantaged accounts. What would happen if he put that discretionary money towards Pay-off or Invest? Nobody on the planet knows that answer, however we can make some educated projections.

The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present. The red line is Frank’s net worth if he puts the $1,500 towards Pay-off. The gray lines are all the outcomes that would have occurred, in the historical period described above, if he had chosen to Invest. The black line is the average of all the gray lines.


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The expected or average of the historical data is $20K higher than the expected result of Pay-off. However, there is a large variance in the outcomes that would have happened if Frank, at some point in the past, decided to Invest instead; he could have ended up $60K worse off by not paying off his mortgage early.

If Frank had decided to Invest at a date in the past in the data set, he would be riding one of those gray lines at random, which could have worked out great for him or been a swift kick to the backside. However, if he had payed off the mortgage early he would have selected the red line and would know exactly the track he was on and where he would end up.

Another interest way to look at the data; Under invest Frank would have a net worth of $0 between year 4 and year 10 while under Pay-off he would reach this mark at 7 years.

This graph emphasizes the actual outcomes of Invest if employed in the past. However, it is not accurate to assume that Frank’s expected value would be greater by $20k than Pay-off if he started in 2017. This statement would assume that the next eight years of stock market returns will be no better or worse than the last 145 years and the probability of any given rate of return occurring is exactly proportion to what occurred during the last 145 years. While this is possible it is very unlikely.

Stock returns in theory are equal to the risk-free rate plus an equity risk premium. Given that the risk-free rate is 0.80%, hence the low mortgage rates, while the normal risk-free rate has been around 4% is it reasonable to assume stock prices are more likely to under-perform historical averages than out-perform? Vanguard seems to think so.


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2017 Vanguard Market Outlook

I have not covered the other factors that influence this decision such as liquidity, asset protection, or psychology which I may in a future post. The purpose of this post was to evaluate the math. In conclusion, the choice between Pay-off and Invest is a risk-reward trade off.

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #1 on: April 13, 2017, 09:00:00 PM »
Very interesting.  You took a 30 year mortgage example, and applied a 7 year period analysis to the situation?  I'm not understanding this:  The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present.  Is it a 7 year rolling period of returns?

A few thoughts, assuming I interpreted your statement and graph correctly:

1. Why not apply the same period of time as the mortgage?  ie for a 30 year mortgage, provide a 30 year period of returns. A 15 year mortgage, 15  years of returns.

2. Does the math include any potential tax savings from the interest deduction, by not paying off the mortgage and investing it?

3. When you include risk in the calculation... is there a way to model the risk of the house depreciating / going underwater and having that cash locked in it as well?

I guess overall this doesn't make sense to me.  Why apply a 7 or 8 year period of returns for a long term mortgage that is 15 or 30 years? 

And the central question here is, will the stock market outperform a 4% fixed rate, tax deductible mortgage over the same period, ie 15 or 30 years?  If the stock market will not outperform a low fixed rate mortgage, then this country has some serious problems ahead of it, imho.

« Last Edit: April 13, 2017, 09:05:17 PM by Valhalla »

JLee

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Re: Mr. Math and paying off your mortgage
« Reply #2 on: April 13, 2017, 09:07:48 PM »
Very interesting.  You took a 30 year mortgage example, and applied a 7 year period analysis to the situation?  I'm not understanding this:  The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present.  Is it a 7 year rolling period of returns?

A few thoughts, assuming I interpreted your statement and graph correctly:

1. Why not apply the same period of time as the mortgage?  ie for a 30 year mortgage, provide a 30 year period of returns. A 15 year mortgage, 15  years of returns.

2. Does the math include any potential tax savings from the interest deduction, by not paying off the mortgage and investing it?

3. When you include risk in the calculation... is there a way to model the risk of the house depreciating / going underwater and having that cash locked in it as well?

I guess overall this doesn't make sense to me. Why apply a 7 or 8 year period of returns for a long term mortgage that is 15 or 30 years? 

And the central question here is, will the stock market outperform a 4% fixed rate, tax deductible mortgage over the same period, ie 15 or 30 years?  If the stock market will not outperform a low fixed rate mortgage, then this country has some serious problems ahead of it, imho.

I'd wager a guess that if you apply 30 year return periods, investing will show a decisive mathematical victory.

bobechs

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Re: Mr. Math and paying off your mortgage
« Reply #3 on: April 13, 2017, 09:16:16 PM »
Maybe it's just me and redounds to my discredit--

BUT-  whenever I read yet another post that conflates the words principle and principal, treating them as interchangable, I suddenly lose the energy to go on with trying to figure out what the poster is mainly driving at.

It's kind of a kryptonite thing, not that I am any kind of Super-being.

Sorry....

marty998

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Re: Mr. Math and paying off your mortgage
« Reply #4 on: April 13, 2017, 09:19:37 PM »
Very interesting.  You took a 30 year mortgage example, and applied a 7 year period analysis to the situation?  I'm not understanding this:  The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present.  Is it a 7 year rolling period of returns?

A few thoughts, assuming I interpreted your statement and graph correctly:

1. Why not apply the same period of time as the mortgage?  ie for a 30 year mortgage, provide a 30 year period of returns. A 15 year mortgage, 15  years of returns.

2. Does the math include any potential tax savings from the interest deduction, by not paying off the mortgage and investing it?

3. When you include risk in the calculation... is there a way to model the risk of the house depreciating / going underwater and having that cash locked in it as well?

I guess overall this doesn't make sense to me. Why apply a 7 or 8 year period of returns for a long term mortgage that is 15 or 30 years? 

And the central question here is, will the stock market outperform a 4% fixed rate, tax deductible mortgage over the same period, ie 15 or 30 years?  If the stock market will not outperform a low fixed rate mortgage, then this country has some serious problems ahead of it, imho.

I'd wager a guess that if you apply 30 year return periods, investing will show a decisive mathematical victory.

Maybe it's because when you pay extra into a mortgage, it no longer runs for 15-30 years. It could very well be paid off after 7 years, at which point you'd put all excess into the market anyway.

Therefore it's only the 7 years of paying extra into the mortgage that matter for the purpose of the analysis?

MDM

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Re: Mr. Math and paying off your mortgage
« Reply #5 on: April 13, 2017, 09:30:04 PM »
Nice analysis!  I'm surprised there are quite a few 30 year periods where the market failed to beat the 4% return of the mortgage payoff.
Based on http://allfinancialmatters.com/2015/12/15/sp-500-30-year-rolling-total-returns/, there are none.  Don't know if that is true if one does a day-by-day analysis of all 30 year periods.

KMMK

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Re: Mr. Math and paying off your mortgage
« Reply #6 on: April 13, 2017, 09:32:33 PM »
Correct me if I'm wrong, but it looks like you are assuming that the house will always increase in value.
The stock market can go up and down, as well as house prices. I don't think it's correct to average historic stock market returns and not average increases or decreases in housing prices.

FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #7 on: April 13, 2017, 09:55:40 PM »


Very interesting.  You took a 30 year mortgage example, and applied a 7 year period analysis to the situation?  I'm not understanding this:  The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present.  Is it a 7 year rolling period of returns?

A few thoughts, assuming I interpreted your statement and graph correctly:

1. Why not apply the same period of time as the mortgage?  ie for a 30 year mortgage, provide a 30 year period of returns. A 15 year mortgage, 15  years of returns.

2. Does the math include any potential tax savings from the interest deduction, by not paying off the mortgage and investing it?

3. When you include risk in the calculation... is there a way to model the risk of the house depreciating / going underwater and having that cash locked in it as well?

I guess overall this doesn't make sense to me.  Why apply a 7 or 8 year period of returns for a long term mortgage that is 15 or 30 years? 

And the central question here is, will the stock market outperform a 4% fixed rate, tax deductible mortgage over the same period, ie 15 or 30 years?  If the stock market will not outperform a low fixed rate mortgage, then this country has some serious problems ahead of it, imho.

OP,  answers to the above post would be appreciated.

I also don't see why you would apply a 7 year time frame to this analysis when comparing it to leveraging a debt with a 30 year term.....

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FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #8 on: April 13, 2017, 10:17:04 PM »
Ok I get it now, the example mortgage is paid off in 7 years with the extra payments, so a 7 year rolling window is used to compare.  That makes sense now - it shows that on average, 7 years in you'd have a paid off house, but your stock account is 20k less than what you would have had if you invested instead.

Quick correction to your assumption.

If you were to pre pay, your stock account would not have 20k less.  Your net worth would be 20k less.  But it would be tied up in the house.

So at the end of 7 years you would have a paid off house but, if you were to invest, that money would now be in the stock market instead of a house.

This makes a huge difference in your long term net worth growth because if you were to invest you would have a "jump start" because you are essentially front loading your portfolio by holding the mortgage.

That front loaded amount then compounds at a much higher rate, because it's a larger pile of investments.

In order to do this calculation correctly you would need to run two scenarios and base them off of an entire 30 year term.

The first scenario Frank would invest the 1500 extra in the stock market for the entire 30 year period dollar cost averaging his way in.

The second scenario Frank would have no money going towards investments for 7 years.  He would then invest 1500 plus the principal/interest payments he use to be paying for the next 23 years.

In both scenarios Frank would have a paid off house. But the real question is in what scenario would Frank have a larger portfolio at the end?



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« Last Edit: April 13, 2017, 10:31:20 PM by FrozenBits »

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #9 on: April 13, 2017, 10:26:24 PM »
^^^ Indeed that is the calculation I'd like to see.

FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #10 on: April 13, 2017, 10:54:45 PM »
Nice graphs.  Curious where you are getting this data from?  Is it extracted from an xcel spreadsheet?  I'd love to get my hands on the spreadsheet if it is and run my own set of different scenarios.

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JLee

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Re: Mr. Math and paying off your mortgage
« Reply #11 on: April 13, 2017, 11:01:06 PM »


Very interesting.  You took a 30 year mortgage example, and applied a 7 year period analysis to the situation?  I'm not understanding this:  The graph below was based on the above assumptions and stock market returns for rolling periods from 1870 to Present.  Is it a 7 year rolling period of returns?

A few thoughts, assuming I interpreted your statement and graph correctly:

1. Why not apply the same period of time as the mortgage?  ie for a 30 year mortgage, provide a 30 year period of returns. A 15 year mortgage, 15  years of returns.

2. Does the math include any potential tax savings from the interest deduction, by not paying off the mortgage and investing it?

3. When you include risk in the calculation... is there a way to model the risk of the house depreciating / going underwater and having that cash locked in it as well?

I guess overall this doesn't make sense to me.  Why apply a 7 or 8 year period of returns for a long term mortgage that is 15 or 30 years? 

And the central question here is, will the stock market outperform a 4% fixed rate, tax deductible mortgage over the same period, ie 15 or 30 years?  If the stock market will not outperform a low fixed rate mortgage, then this country has some serious problems ahead of it, imho.

OP,  answers to the above post would be appreciated.

I also don't see why you would apply a 7 year time frame to this analysis when comparing it to leveraging a debt with a 30 year term.....

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Sorry been busy with a few other things.

1. i am using rolling 30 stock market returns. See other posts for the scale reasons and larger periods examined.

2. No tax savings assumed. $165,000 * 4.00% =$6,600, far below the standard deduction. Frank is a cranky SOB and does not give enough to in combination with his mortgage to get over the standard deduction hurdle. Oh, and he is married.

3. The house going under can happen with or without a mortgage and is more likely to happen with a mortgage. As a mortgage is paid down you could re-borrow the money with a HELOC.

For tax year 2016: $4050 exemption + $4050 exemption + $6600 interest = $14,700 (plus property taxes).  The MFJ standard deduction is $12,600.

This claim makes me doubt the validity of the rest of the math.

Edit: I'm wrong. Don't mind me.
« Last Edit: April 13, 2017, 11:16:27 PM by JLee »

MDM

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Re: Mr. Math and paying off your mortgage
« Reply #12 on: April 13, 2017, 11:06:22 PM »
For tax year 2016: $4050 exemption + $4050 exemption + $6600 interest = $14,700 (plus property taxes).  The MFJ standard deduction is $12,600.

This claim makes me doubt the validity of the rest of the math.
While it is true that 4050*2 + 6600 = 14700, exemptions are separate from deductions.  See 2016 Form 1040.

FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #13 on: April 13, 2017, 11:07:28 PM »
Nice graphs.  Curious where you are getting this data from?  Is it extracted from an xcel spreadsheet?  I'd love to get my hands on the spreadsheet if it is and run my own set of different scenarios.

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Yep, it is based on a excel spreadsheet. I will have to clean it up and post it.
Please do!  I'd love to mess around with this :)

Maybe do a Google Doc share?

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JLee

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Re: Mr. Math and paying off your mortgage
« Reply #14 on: April 13, 2017, 11:09:58 PM »
For tax year 2016: $4050 exemption + $4050 exemption + $6600 interest = $14,700 (plus property taxes).  The MFJ standard deduction is $12,600.

This claim makes me doubt the validity of the rest of the math.
While it is true that 4050*2 + 6600 = 14700, exemptions are separate from deductions.  See 2016 Form 1040.

Quote
42 Exemptions. If line 38 is $155,650 or less, multiply $4,050 by the number on line 6d. Otherwise, see instructions
If AGI is under $155,650, is the end result not the same?

Edit: Ahh I see, they are itemized separately. My bad. : )

I easily clear the standard deduction every year with a $120k mortgage at 3.75%, fwiw...I am single, though.
« Last Edit: April 13, 2017, 11:11:39 PM by JLee »

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #15 on: April 13, 2017, 11:15:39 PM »
Nice plot for the 30 year. Can you show the average trend lines? Thanks.

FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #16 on: April 13, 2017, 11:47:53 PM »
Nice plot for the 30 year. Can you show the average trend lines? Thanks.
Ah yes, average trend would be great.

Also, it looks like you are tracking the house as a debt in this calculation.

I think a simpler way to project this scenario would be to start with just looking at the investment portfolio which would start at $0. 

When comparing this scenario we can basically ignore the mortgage for the most part.  The only thing of significance is the amatorization schedule shift that will be observed by the pre pay scenario.

The pre pay scenario would then be a 0 on your chart until after 7 years when the mortgage is paid off.  Then years 8 through 30 the pre pay frank would be investing 2,287.74 (1500+787.74 of mortgage P/I) per month into his brokerage account.

The invest Frank would be investing 1,500.00 over the entire 30 year period.

I'd love to see a graph that shows just the portfolio of pre pay Frank trying to catch up to invest first Frank over years 0 to 30.

Excited to get my hands on this excel spreadsheet. So many different scenarios to run! :D

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rpr

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Re: Mr. Math and paying off your mortgage
« Reply #17 on: April 14, 2017, 12:39:00 AM »
I made a slightly different plot based of request by FrozenBits.



As you can see, Invest beats Prepay over every single 30 year period starting at the beginning of years 1928 to 1987.

Data is from: http://pages.stern.nyu.edu/~adamodar/New_Home_Page/datafile/histretSP.html

Edit: Figure is updated. I needed the numbers to be multiplied by 12 as the investment calcuations were made yearly.
« Last Edit: April 14, 2017, 09:35:34 AM by rpr »

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #18 on: April 14, 2017, 01:06:35 AM »


1. i am using rolling 30 stock market returns. See other posts for the scale reasons and larger periods examined.

2. No tax savings assumed. $165,000 * 4.00% =$6,600, far below the standard deduction. Frank is a cranky SOB and does not give enough to in combination with his mortgage to get over the standard deduction hurdle. Oh, and he is married.

3. The house going under can happen with or without a mortgage and is more likely to happen with a mortgage. As a mortgage is paid down you could re-borrow the money with a HELOC.

2. itemized deduction is often superior because you can itemize 2 of the biggest items - mortgage interest, and state / local taxes (income tax, property tax, etc).  Depending on how much Frank and his wife earn, and how much they pay in state taxes, it could result in a nice deduction from Federal income.

3. HELOC, assuming one could obtain it when economy is doing poor and housing prices are depressed, would be much more expensive and difficult to obtain.  Especially if one is FIRE and has no job, will have to show good cash flow to obtain HELOC, which begs the question of why they need a loan in the first place if they have good cash flow.   So it's a catch 22, for someone who is heavily invested in their house and don't have enough liquidity / investments to help meet emergency needs during down times.

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #19 on: April 14, 2017, 01:19:34 AM »


Interestingly enough, out of both scenarios and all rolling periods, Frank ends with the most money starting in year 1970 with the strategy of.......

drum roll please....

Paying off the mortgage early!

I'm curious how this was calculated.

In 1970, the 30 year fixed rate was 7.25%.   So to beat that, the stock market had to do better than that. 

http://www.doctorhousingbubble.com/federal-reserve-fighting-inflation-in-the-1970s-and-restraining-the-housing-market-today-the-federal-reserve-is-juicing-the-housing-market-trying-to-cause-inflation-researching-the-1970s-and-1980s/

http://www.stockpickssystem.com/historical-rate-of-return/

The stock market had 5.82% return in the 70's, 17.57% in the 80's, and 18.17% in the 90's, for an average return of 13.85% over the 3 decades.  How could the mortgage rate of 7.25% beat 13.85%?


rpr

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Re: Mr. Math and paying off your mortgage
« Reply #20 on: April 14, 2017, 01:23:00 AM »


Interestingly enough, out of both scenarios and all rolling periods, Frank ends with the most money starting in year 1970 with the strategy of.......

drum roll please....

Paying off the mortgage early!


I'm curious how this was calculated.

In 1970, the 30 year fixed rate was 7.25%.   So to beat that, the stock market had to do better than that. 

http://www.doctorhousingbubble.com/federal-reserve-fighting-inflation-in-the-1970s-and-restraining-the-housing-market-today-the-federal-reserve-is-juicing-the-housing-market-trying-to-cause-inflation-researching-the-1970s-and-1980s/

http://www.stockpickssystem.com/historical-rate-of-return/

The stock market had 5.82% return in the 70's, 17.57% in the 80's, and 18.17% in the 90's, for an average return of 13.85% over the 3 decades.  How could the mortgage rate of 7.25% beat 13.85%?

Indeed, my plot shows that investing beats prepay by almost $200K starting in 1970. Plus 1970's 30 year period includes the tech late 1990s peak.

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #21 on: April 14, 2017, 01:26:48 AM »


Interestingly enough, out of both scenarios and all rolling periods, Frank ends with the most money starting in year 1970 with the strategy of.......

drum roll please....

Paying off the mortgage early!


I'm curious how this was calculated.

In 1970, the 30 year fixed rate was 7.25%.   So to beat that, the stock market had to do better than that. 

http://www.doctorhousingbubble.com/federal-reserve-fighting-inflation-in-the-1970s-and-restraining-the-housing-market-today-the-federal-reserve-is-juicing-the-housing-market-trying-to-cause-inflation-researching-the-1970s-and-1980s/

http://www.stockpickssystem.com/historical-rate-of-return/

The stock market had 5.82% return in the 70's, 17.57% in the 80's, and 18.17% in the 90's, for an average return of 13.85% over the 3 decades.  How could the mortgage rate of 7.25% beat 13.85%?

Indeed, my plot shows that investing beats prepay by almost $200K starting in 1970. Plus 1970's 30 year period includes the tech late 1990s peak.
Thanks for running the numbers and the chart, that was what I was expecting to see.  I would have assumed if someone obtained a mortgage in 1980 with a rate of almost 20%, it would have made more sense to pay off that mortgage than invest, as the stock market returns were far lower than 20% after that period of time.  I just don't understand how the math would work starting in 1970 when the mortgage rate was only 7.25%.

rantk81

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Re: Mr. Math and paying off your mortgage
« Reply #22 on: April 14, 2017, 07:40:52 AM »
At what point do we need a new forum sub-group for all the threads about early mortgage payoff? :P

brooklynguy

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Re: Mr. Math and paying off your mortgage
« Reply #23 on: April 14, 2017, 07:56:52 AM »
In order to do this calculation correctly you would need to run two scenarios and base them off of an entire 30 year term.

The first scenario Frank would invest the 1500 extra in the stock market for the entire 30 year period dollar cost averaging his way in.

The second scenario Frank would have no money going towards investments for 7 years.  He would then invest 1500 plus the principal/interest payments he use to be paying for the next 23 years.

In both scenarios Frank would have a paid off house. But the real question is in what scenario would Frank have a larger portfolio at the end?

I just ran these two scenarios in cFIREsim.  It reports, contrary to Virtus' claim, that there was no historical period in which the mortgage prepayment strategy would have come out ahead.  (Once Virtus publishes his/her data, we can try to compare to cFIREsim's data and assumptions to determine the source of the discrepancy.)  According to cFIREsim, the absolute worst historical case for the "no mortgage prepayment" scenario resulted in a final portfolio value that exceeded the final portfolio value of the absolute worst historical case for the "mortgage prepayment" scenario by more than $132K.

Results

In the "mortgage prepayment" scenario, the portfolio ending value statistics in cFIREsim were as follows:

Average:  $  1,181,218
Median:    $     967,525
Highest:   $  2,815,498
Lowest:    $     433,935

In the "no mortgage prepayment" scenario, the portfolio ending value statistics in cFIREsim were as follows:

Average:  $  1,476,351
Median:    $  1,330,471
Highest:   $  3,267,481
Lowest:    $     565,958

Methodology

I used the following methodology in running the cFIREsim backtests:

In both scenarios, I set the investment allocation to 100% equities, left the invest fees at the default setting of 0.18%, and set the initial yearly spending to zero.

In the "mortgage prepayment" scenario, I added an "other income" line item of $27,456 commencing in year 2024, and set it as non-inflation adjusted.  This represents the annual amount of funds that would be plowed into investments under this scenario (an amount per month equal to $788 (principal + interest) + $1500) commencing when the mortgage loan is paid off after seven years.

In the "no mortgage prepayment" scenario, I added an "other income" line item of $18,000 commencing immediately (in year 2017), and set it as non-inflation adjusted.  This represents the annual amount of funds that would be plowed into investments under this scenario ($1500 per month) from the get-go.

HenryDavid

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Re: Mr. Math and paying off your mortgage
« Reply #24 on: April 14, 2017, 08:00:47 AM »
Oh, you Americans and your mortgage-interest tax deductions.
Most countries do not do this. Certainly not Canada.
This helps tilt many of us up north toward paying off the damn mortgage. (Even if your deduction mainly helps millionaires.)

One other factor that propels people toward doing this is the painful awareness that our parents faced 18-10% mortgage interest in the 1980s.
So recent low rates have been seen as a historic window of opportunity to get mortgages paid down. If not entirely off.

ReadySetMillionaire

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Re: Mr. Math and paying off your mortgage
« Reply #25 on: April 14, 2017, 08:48:08 AM »
What about pre-paying the mortgage to get rid of PMI?  My amortization schedule shows I'll be paying approximately $56/month in PMI for about nine years ($672/year, or $6,048 total). I've checked with my lender and they will remove PMI once we hit 80% equity.

I figure getting rid of the PMI in itself adds to my rate of return in pre-paying the mortgage. In other words, getting rid of PMI adds to the rate of return on pre-paying the mortgage.

Any analysis on this?  This seems to be a good balance--prepay only to the extent to get rid of PMI, then let it ride, which conveniently enough will coincide closely with our FIRE date.

Scortius

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Re: Mr. Math and paying off your mortgage
« Reply #26 on: April 14, 2017, 09:26:26 AM »
What about pre-paying the mortgage to get rid of PMI?  My amortization schedule shows I'll be paying approximately $56/month in PMI for about nine years ($672/year, or $6,048 total). I've checked with my lender and they will remove PMI once we hit 80% equity.

I figure getting rid of the PMI in itself adds to my rate of return in pre-paying the mortgage. In other words, getting rid of PMI adds to the rate of return on pre-paying the mortgage.

Any analysis on this?  This seems to be a good balance--prepay only to the extent to get rid of PMI, then let it ride, which conveniently enough will coincide closely with our FIRE date.

It's not too tough, you can calculate a schedule of pre-payment to get you to 20% equity (I assume you meant 80% remaining).  At that point, look at how much extra you spent in accelerated payments.  Calculate that compared to the total amount of mortgage interest that you don't have to pay due to those payments, plus the $6,048 (minus the PMI that you did pay during that period).  That will give you your rate of return on your accelerated payment schedule.  It may be worth it, it may not, depending on your interest rate and how quickly you can hit 20%.

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #27 on: April 14, 2017, 09:31:06 AM »

One other factor that propels people toward doing this is the painful awareness that our parents faced 18-10% mortgage interest in the 1980s.
So recent low rates have been seen as a historic window of opportunity to get mortgages paid down. If not entirely off.
I would have come to the opposite conclusion. Funny :)

runewell

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Re: Mr. Math and paying off your mortgage
« Reply #28 on: April 14, 2017, 09:36:17 AM »
What about pre-paying the mortgage to get rid of PMI?  My amortization schedule shows I'll be paying approximately $56/month in PMI for about nine years ($672/year, or $6,048 total). I've checked with my lender and they will remove PMI once we hit 80% equity.

I figure getting rid of the PMI in itself adds to my rate of return in pre-paying the mortgage. In other words, getting rid of PMI adds to the rate of return on pre-paying the mortgage.

Any analysis on this?  This seems to be a good balance--prepay only to the extent to get rid of PMI, then let it ride, which conveniently enough will coincide closely with our FIRE date.

The PMI question is more complicated.  When I bought my first house with 3% down in 1998, there was quick appereciation and the PMI was gone in two years.  One variable is the house appreciation (or rather than a stochastic approach you could set appreciation as some fixed value X% and calculate the amount of time you would expect to pay PMI).

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #29 on: April 14, 2017, 10:07:06 AM »


Interestingly enough, out of both scenarios and all rolling periods, Frank ends with the most money starting in year 1970 with the strategy of.......

drum roll please....

Paying off the mortgage early!


I'm curious how this was calculated.

In 1970, the 30 year fixed rate was 7.25%.   So to beat that, the stock market had to do better than that. 

http://www.doctorhousingbubble.com/federal-reserve-fighting-inflation-in-the-1970s-and-restraining-the-housing-market-today-the-federal-reserve-is-juicing-the-housing-market-trying-to-cause-inflation-researching-the-1970s-and-1980s/

http://www.stockpickssystem.com/historical-rate-of-return/

The stock market had 5.82% return in the 70's, 17.57% in the 80's, and 18.17% in the 90's, for an average return of 13.85% over the 3 decades.  How could the mortgage rate of 7.25% beat 13.85%?

Indeed, my plot shows that investing beats prepay by almost $200K starting in 1970. Plus 1970's 30 year period includes the tech late 1990s peak.
Thanks for running the numbers and the chart, that was what I was expecting to see.  I would have assumed if someone obtained a mortgage in 1980 with a rate of almost 20%, it would have made more sense to pay off that mortgage than invest, as the stock market returns were far lower than 20% after that period of time.  I just don't understand how the math would work starting in 1970 when the mortgage rate was only 7.25%.

The analysis does not use differing mortgage rates. It uses the same mortgage rate of 4% in all scenarios.
In this case, it makes paying off the mortgage even less worthwhile, compared to the average stock market return of 13.85% from 1970 - 2000.    As Brooklynguy noted, it would be good to see your exact underlying data to understand the source of the data, and how you did the calculations.
« Last Edit: April 14, 2017, 10:10:04 AM by Valhalla »

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #30 on: April 14, 2017, 10:08:23 AM »

One other factor that propels people toward doing this is the painful awareness that our parents faced 18-10% mortgage interest in the 1980s.
So recent low rates have been seen as a historic window of opportunity to get mortgages paid down. If not entirely off.
I would have come to the opposite conclusion. Funny :)
Exactly.  If I had a 18 - 20% mortgage, I would be paying that puppy off as fast as I can, or refinance every time the interest rate dropped 2 - 3% to step down the interest rate cost.  At current 4% mortgage rate levels, I'd ride that puppy out as long as I can.

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #31 on: April 14, 2017, 10:15:05 AM »
Virtus -- Looks like you and I have a significant difference in our results for 30 year networths when we compare from 1928 onwards.


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Re: Mr. Math and paying off your mortgage
« Reply #32 on: April 14, 2017, 10:20:59 AM »
-The calculations do include taxes on long-term capital gains/dividends at a rate of 15%. All gains and losses are realized in the year they are recognized. Yes this is an oversimplification because gains can be deferred and only $3,000 of losses can be recognized against regular income in any year. At some point you have to make simplifications to make the model creation not become a full-time job. Further even deferred gains will be recognized at some point in the future and thus represent a deferred tax liability.

This assumption creates an enormous drag on investment performance that won't exist in the real world, and might go a long way toward explaining why your results differ so drastically from both my results in reply # 30 (which used cFIREsim and its underlying data sources and assumptions) and rpr's results in reply # 24 (which used FRED data and unspecified assumptions), both of which indicated that there was never a historical 30-year period in which Frank's "pay-off" strategy would have outperformed Frank's "invest" strategy.

While it might be justifiable to assume for simplicity's sake that unrealized capital gains will eventually be taxed upon disposition (though, as mustachians know or ought to know, even that won't necessarily be true in the real world), I don't think it is justifiable to assume that unrealized capital gains will be taxed as if they were realized.  That creates totally unrealistic tax drag on investment performance, making the model depart from reality to a much greater extent than simply ignoring tax effects altogether would have done.

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #33 on: April 14, 2017, 10:32:10 AM »
brooklynguy -- Thanks for providing a possible explanation.

Virtus -- I'd be curious if you could turn off impact of taxes on your model. Just want to see if we get similar answers then. Thanks.

Cwadda

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Re: Mr. Math and paying off your mortgage
« Reply #34 on: April 14, 2017, 10:43:54 AM »


In all seriousness though, the whole mortgage payoff deal has lead to some very interesting discussions.

FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #35 on: April 14, 2017, 11:21:33 AM »
In order to do this calculation correctly you would need to run two scenarios and base them off of an entire 30 year term.

The first scenario Frank would invest the 1500 extra in the stock market for the entire 30 year period dollar cost averaging his way in.

The second scenario Frank would have no money going towards investments for 7 years.  He would then invest 1500 plus the principal/interest payments he use to be paying for the next 23 years.

In both scenarios Frank would have a paid off house. But the real question is in what scenario would Frank have a larger portfolio at the end?


Average:  $  1,181,218
Median:    $     967,525
Highest:   $  2,815,498
Lowest:    $     433,935

In the "no mortgage prepayment" scenario, the portfolio ending value statistics in cFIREsim were as follows:

Average:  $  1,476,351
Median:    $  1,330,471
Highest:   $  3,267,481
Lowest:    $     565,958



Yep, this is what I would expect to see, and what I have seen when running the numbers in the past.  Worst case scenario invest beats pre pay by 132k, average beats pre-pay by 295k, and highest is 452k.

Now, the best chance for pre-pay Frank to actually come out ahead is if he shortens his time to mortgage payoff.  If he pays the home off in 2 years it is possible that he could surpass invest Frank, but only if invest Frank has the market tank on during years 1-2.  Talk about worse case scenario.

moof

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Re: Mr. Math and paying off your mortgage
« Reply #36 on: April 14, 2017, 12:15:54 PM »
Out of curiosity I ran my situation.  $260k mortgage with ~22k annual payments for 15 years, 3.0% interest rate.  We went for a 15 years when we refinanced down from 4.75% to split the difference so to speak, and to keep payments about the same.

Net present value today of those payments in $351k earning 3%with a 2% inflation rate.  Firecalc says my inflation adjusted investment sticking with my current 80/20 portfolio could range from ~250k to 1.1M, average of $620k.  Investing won 81% of the time under this scenario without even talking about deductions.

I'm pretty happy sticking with my current plan of just paying it off over 15 years.  Odds are decidedly in my favor, and I'd rather increase the odds of being able to retire better and/or sooner with a modest 20% risk that if we get a historically bad 15 years of returns that I might have to work another year or two due to my choice.
« Last Edit: April 14, 2017, 02:07:20 PM by moof »

Bateaux

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Re: Mr. Math and paying off your mortgage
« Reply #37 on: April 14, 2017, 12:22:56 PM »
Glad we just bought a cheap house and never moved up.  Mortgage paid off in just a few years and more for investment.

brooklynguy

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Re: Mr. Math and paying off your mortgage
« Reply #38 on: April 14, 2017, 12:25:37 PM »
It is not that far away from reality. Dividends and capital gains realized through turnover will be recognized when realized. Capital gains that are deferred will  be realized at a latter date.

But there is no turnover whatsoever for the entirety of the 30-year period (during which stocks are bought, but none are sold).  I can understand the tax assumption on dividends, but not on unrealized capital gains (which represent the lion's share of the total return).  So artificially assuming tax liability on unrealized capital gains creates an unrealistic and significant drag on investment performance.  Even if you assume that the capital gains will be taxed upon disposition after the 30-year period, the tax-deferred growth during that period provides a significant performance boost.

But it doesn't look like the tax assumptions were the sole source of the discrepancies between your results and mine and rpr's, because your new chart with 0% assumed taxes still shows the "payoff" alternative outperforming in some cases (while in my and rpr's backtests it never did).

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #39 on: April 14, 2017, 12:46:18 PM »


Interestingly enough, out of both scenarios and all rolling periods, Frank ends with the most money starting in year 1970 with the strategy of.......

drum roll please....

Paying off the mortgage early!


I'm curious how this was calculated.

In 1970, the 30 year fixed rate was 7.25%.   So to beat that, the stock market had to do better than that. 

http://www.doctorhousingbubble.com/federal-reserve-fighting-inflation-in-the-1970s-and-restraining-the-housing-market-today-the-federal-reserve-is-juicing-the-housing-market-trying-to-cause-inflation-researching-the-1970s-and-1980s/

http://www.stockpickssystem.com/historical-rate-of-return/

The stock market had 5.82% return in the 70's, 17.57% in the 80's, and 18.17% in the 90's, for an average return of 13.85% over the 3 decades.  How could the mortgage rate of 7.25% beat 13.85%?

Indeed, my plot shows that investing beats prepay by almost $200K starting in 1970. Plus 1970's 30 year period includes the tech late 1990s peak.
Thanks for running the numbers and the chart, that was what I was expecting to see.  I would have assumed if someone obtained a mortgage in 1980 with a rate of almost 20%, it would have made more sense to pay off that mortgage than invest, as the stock market returns were far lower than 20% after that period of time.  I just don't understand how the math would work starting in 1970 when the mortgage rate was only 7.25%.

The analysis does not use differing mortgage rates. It uses the same mortgage rate of 4% in all scenarios.
In this case, it makes paying off the mortgage even less worthwhile, compared to the average stock market return of 13.85% from 1970 - 2000.    As Brooklynguy noted, it would be good to see your exact underlying data to understand the source of the data, and how you did the calculations.

You are making the mistake of looking at averages and forgetting about volatility. If frank chose the invest strategy, his investments would be hammered in 1973 and 1974. If he follows the Pay-off strategy, his house payment would disappear in 1977,allowing him to invest more per year than the Invest strategy over the next 10 years of strong returns.

            Nominal/ Real
1970     4%            -2%
1971   15%      11%
1972   19%            15%
1973       -15%   -22%
1974        -27%       -35%

All returns were sourced from: www.moneychimp.com
So in short, if you believe the market will tank, and have double digit negative losses, it is a no brainer to stay the heck out of the market.  Keep the money in cash, or pay off the mortgage, and that will beat the investing strategy (almost everything will beat investing, in this case, including buying beanie babies), assuming you cash out of the losses as well.

So only in a very specific, rare, circumstance, was your scenario able to beat the investing strategy.  So unless we have the exact same situation (prepay your mortgage over a 7 year period, during which the market tanks), which is highly improbable, basically prepay never beats investing.

Thanks for the explanation.  It seems you almost have to mock up a situation deliberately for the prepay strategy to come out ahead, whereas in 99.99% of all other scenarios (basically staying out of the market during some of the worst years), the investing strategy beats prepay.

If we all could predict when the worst years of the market would be, we would be rich beyond compare.
« Last Edit: April 14, 2017, 12:53:14 PM by Valhalla »

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #40 on: April 14, 2017, 12:50:30 PM »
I made an updated plot based on the moneychimp data link (http://www.moneychimp.com/features/market_cagr.htm) provided by virtus.



In all except one 30 year period (1925), invest beat prepay. We still have the discrepancy.

PS: If anyone is interested, I'd be happy to share my google sheet via PM.

mizzourah2006

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Re: Mr. Math and paying off your mortgage
« Reply #41 on: April 14, 2017, 01:05:26 PM »
It is not that far away from reality. Dividends and capital gains realized through turnover will be recognized when realized. Capital gains that are deferred will  be realized at a latter date.

But there is no turnover whatsoever for the entirety of the 30-year period (during which stocks are bought, but none are sold).  I can understand the tax assumption on dividends, but not on unrealized capital gains (which represent the lion's share of the total return).  So artificially assuming tax liability on unrealized capital gains creates an unrealistic and significant drag on investment performance.  Even if you assume that the capital gains will be taxed upon disposition after the 30-year period, the tax-deferred growth during that period provides a significant performance boost.

But it doesn't look like the tax assumptions were the sole source of the discrepancies between your results and mine and rpr's, because your new chart with 0% assumed taxes still shows the "payoff" alternative outperforming in some cases (while in my and rpr's backtests it never did).

Capital gains are taxed not only upon sale but also turnover. Vanguard's turnover is low in most funds but between dividends, turnover, and the taxes upon liquidate including taxes in the calculation produces a more accurate result. 

https://www.fidelity.com/tax-information/tax-topics/mutual-funds

So one other assumption is that Fred and Wife are maxing out both of their 401ks, IRAs and HSAs as well I assume.

Valhalla

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Re: Mr. Math and paying off your mortgage
« Reply #42 on: April 14, 2017, 01:21:59 PM »
So in short, if you believe the market will tank, and have double digit negative losses, it is a no brainer to stay the heck out of the market.  Keep the money in cash, or pay off the mortgage, and that will beat the investing strategy (almost everything will beat investing, in this case, including buying beanie babies), assuming you cash out of the losses as well.

So only in a very specific, rare, circumstance, was your scenario able to beat the investing strategy.  So unless we have the exact same situation (prepay your mortgage over a year period, during which the market tanks), which is highly improbable, basically prepay never beats investing.

Thanks for the explanation.  It seems you almost have to mock up a situation deliberately for the prepay strategy to come out ahead, whereas in 99.99% of all other scenarios (basically staying out of the market during some of the worst years), the investing strategy beats prepay.

If we all could predict when the worst years of the market would be, we would be rich beyond compare.

In short, every time one of those little orange dots is above the little blue dots you would have been better off, starting at that year in history, paying off your mortgage rather than investing. The data I provided was one time the little orange dot was above the blue dot. Are 99.99% of the blue dots above the orange dots? Nope. If you look at the various graphs there are many times the orange dots are above the blue dots and vice versa.
so in short, if you're on pace to pay off your mortgage in 7 years, instead of 15 or 30 years, there are a few potential 7 years of volatility during which it would be wiser to stay out of the market during those 7 years, instead of being invested, assuming you're lucky enough to pre-pay the mortgage during those periods (same as market timing essentially).

I'm curious what the hard numbers are, assuming one is lucky enough to time the market at the worst periods and stay out of it while plowing the cash into a mortgage.   What is the delta between the 2 amounts of net worth?

boarder42

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Re: Mr. Math and paying off your mortgage
« Reply #43 on: April 14, 2017, 01:31:53 PM »
So in short, if you believe the market will tank, and have double digit negative losses, it is a no brainer to stay the heck out of the market.  Keep the money in cash, or pay off the mortgage, and that will beat the investing strategy (almost everything will beat investing, in this case, including buying beanie babies), assuming you cash out of the losses as well.

So only in a very specific, rare, circumstance, was your scenario able to beat the investing strategy.  So unless we have the exact same situation (prepay your mortgage over a year period, during which the market tanks), which is highly improbable, basically prepay never beats investing.

Thanks for the explanation.  It seems you almost have to mock up a situation deliberately for the prepay strategy to come out ahead, whereas in 99.99% of all other scenarios (basically staying out of the market during some of the worst years), the investing strategy beats prepay.

If we all could predict when the worst years of the market would be, we would be rich beyond compare.

In short, every time one of those little orange dots is above the little blue dots you would have been better off, starting at that year in history, paying off your mortgage rather than investing. The data I provided was one time the little orange dot was above the blue dot. Are 99.99% of the blue dots above the orange dots? Nope. If you look at the various graphs there are many times the orange dots are above the blue dots and vice versa.
so in short, if you're on pace to pay off your mortgage in 7 years, instead of 15 or 30 years, there are a few potential 7 years of volatility during which it would be wiser to stay out of the market during those 7 years, instead of being invested, assuming you're lucky enough to pre-pay the mortgage during those periods (same as market timing essentially).

I'm curious what the hard numbers are, assuming one is lucky enough to time the market at the worst periods and stay out of it while plowing the cash into a mortgage.   What is the delta between the 2 amounts of net worth?

these few years are coupled with inflation.  inflation further deflates real returns.
« Last Edit: April 14, 2017, 01:35:23 PM by boarder42 »

bacchi

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Re: Mr. Math and paying off your mortgage
« Reply #44 on: April 14, 2017, 01:59:49 PM »
I'm curious what the hard numbers are, assuming one is lucky enough to time the market at the worst periods and stay out of it while plowing the cash into a mortgage.   What is the delta between the 2 amounts of net worth?

Better yet, is there a signal, like a fibonacci retracement on the candle wick?

boarder42

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Re: Mr. Math and paying off your mortgage
« Reply #45 on: April 14, 2017, 02:13:13 PM »
so we've really got

1. market timing at play here
2. grossly over estimated tax consequences when investing vs prepaying
3. a black box of data still that no one can replicate the OPs numbers.

You can manipulate math in any way to make it look like you come out ahead by changing variables but it still doesnt work out based on the common assumptions we make around here for FIRE.

FIreDrill

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Re: Mr. Math and paying off your mortgage
« Reply #46 on: April 14, 2017, 02:25:51 PM »
I made an updated plot based on the moneychimp data link (http://www.moneychimp.com/features/market_cagr.htm) provided by virtus.



In all except one 30 year period (1925), invest beat prepay. We still have the discrepancy.

PS: If anyone is interested, I'd be happy to share my google sheet via PM.

Are you factoring in inflation?
Why are you factoring inflation into this scenario?

Inflation should only be factored in if you have an expense that is affected by it IMO.

If you are factoring in inflation, how are our factoring it in? Just reducing the overall return?

Sent from my SM-G935F using Tapatalk


Scortius

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Re: Mr. Math and paying off your mortgage
« Reply #47 on: April 14, 2017, 03:14:53 PM »
I made an updated plot based on the moneychimp data link (http://www.moneychimp.com/features/market_cagr.htm) provided by virtus.



In all except one 30 year period (1925), invest beat prepay. We still have the discrepancy.

PS: If anyone is interested, I'd be happy to share my google sheet via PM.

Are you factoring in inflation?
Why are you factoring inflation into this scenario?

Inflation should only be factored in if you have an expense that is affected by it IMO.

If you are factoring in inflation, how are our factoring it in? Just reducing the overall return?

Sent from my SM-G935F using Tapatalk

I built in the functionality to the spreadsheet to use real or nominal returns. Including inflation or not including inflation is a complicated topic and depends to some degree on what question you are trying to answer. If the question was what would be the results if Frank chose "Invest" but then paid off the mortgage when he hit a net worth of 0, then inflation should not be included in the analysis on the return side. However, if Frank was looking at the 30 year time range and wanted to know what his terminal net worth would be then it would have to be included.

There are two sides to inflation. The first is on the expense side. If he spends money from his portfolio in the future, it will have less buying power than it will today. The other side is his income should fluctuate with CPI-U.

Ideally when I have the time, I want to build in an inflation adjustment for the income side ($1,500) that uses the historical CPI-U.

In general, high inflation tilts the argument much more strongly towards not paying off a house.  Mortgage payments are fixed, and thus are cheaper to pay off in the future.  In general, the stock market is a reasonable guard against inflation as your equity is in companies that rise in value with inflation (or much higher in economies of rapid growth).  Housing values mirror inflation as well, but you realize that gain regardless of whether or not you pay off the house early.

If you're comparing how much buying power you have at the end of the scenario, it doesn't really matter, since each final value will be scaled by the same amount.

rpr

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Re: Mr. Math and paying off your mortgage
« Reply #48 on: April 14, 2017, 03:19:55 PM »

In general, high inflation tilts the argument much more strongly towards not paying off a house.  Mortgage payments are fixed, and thus are cheaper to pay off in the future.  In general, the stock market is a reasonable guard against inflation as your equity is in companies that rise in value with inflation (or much higher in economies of rapid growth).  Housing values mirror inflation as well, but you realize that gain regardless of whether or not you pay off the house early.

If you're comparing how much buying power you have at the end of the scenario, it doesn't really matter, since each final value will be scaled by the same amount.

Agreed. In fact when carrying a fixed interest debt, deflation would be a lot more worrisome.

boarder42

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Re: Mr. Math and paying off your mortgage
« Reply #49 on: April 14, 2017, 07:50:34 PM »
Correct. But one of the reasons the 70s appear to be better for pay offers in addition to paying insane amounts of taxes is when the inflation is factored into the real returns it makes them look like dog shit. If the market returns 100% and inflation is 200% you have a real return of around -100%. In the real world this helps the guy with a mortgage a shit ton more than the guy with a paid off house but the way op is using real returns it hurts the guy investing a ton. One reason why the 70s appear to be worse for. Investing than they truly were.