20 year or 30 year mortgage?

[crazysteve5575]

Im in the process of purchasing a house and i cant decide if i should get a 20 or 30 year mortgage.

right now a 20 year mortgage will cost us 980 a month(3.635% interest) for P&I  30y is 798 (4.0%int)

so that make the total cost over the term 30years 287,280 and 20y 235,200 thats a savings of 52,080 over the 30 years.

The cost is 182 additional per month.  if i take that 182 a month and put it in retirement over 20 years its 43680 + interest (not sure how to calculate this)

we can afford both, but having the extra 182 to put in investments might be nice.  also paying off the mortgage in 20 years is nice because all that 20-30 year money could be investments as well.

also what if i get the 30, and just pay the 182 extra would i still pay it off in roughly 20 years?  is there a benefit to locking into a 20 year mortgage?

[Jack]

To figure out which is better, you can do a present-value analysis.

Option 1 (30-year mortgage)

PV_{option 1} = PV_mortgage + PV_investment

A_mortgage = -$798
A_investment = $182
i_mortgage = 4%/year / 12 months = 0.003333/month
i_investment = 7%/year (assumed) = 0.005833/month
n_mortgage = 360 months
n_investment = 360 months

PV_mortgage = (-$798 / 0.003333) * (1 - (1/(1+0.003333)³⁶⁰)) = -$167,150.07
PV_investment = ($182 / 0.005833) * (1 - (1/(1+0.005833)³⁶⁰)) = 27,355.98

PV_{option 1} = -$139,794.09

Option 2 (20-year mortgage)

PV_{option 2} = PV_mortgage + PV(FV_{investment at year 20})

Initial:
A_mortgage = -$980
i_mortgage = 2.635%/year = 0.00219583/month
n_mortgage = 240 months

PV_mortgage = (-$980/ 0.00219583) * (1 - (1/(1+0.00219583)²⁴⁰)) = -$182,663.98

From year 20 to year 30:
A_investment = $980
i_investment = 7%/year (assumed)
n_investment = 120 months

FV_{investment at year 20} = ($980/ 0.07) * (1 - (1/(1+0.07)¹²⁰)) = $84,403.83
PV_investment = FV_{investment at year 20} / (1 + 0.07)²⁴⁰ = $20,898.56

PV_{option 2} = -$161,765.42

-$139,794.09 > -$161,765.42, so the 30-year mortgage is better (under the assumption that stock market return is 7% -- otherwise, YMMV).

[Ensign1999]

Jack's analysis is spot on.  Based on a 7% return on the market, the 30 year would be the better choice.  There are other personal decisions to take into account to help make your choice though.

-Will you invest the difference if you go with the 30 year, or will it end up getting spent?  If it might get spent, then maybe the 20 year might be better because it forces you to put the money towards paying down your dept/invest in your future FI by getting the mortgage paid off sooner.
-You say you can afford the extra $182 a month on the 20 year mortgage, but could you if one of you lost your jobs.  Would having the flexibility to choose to pay the extra or not be something that would help you sleep at night?  If so, then maybe the 30 year is better.

These are just a couple of questions that popped into my mind when I read your post.  Each person's situation is different and there is no one right answer when it comes to this.

[crazysteve5575]

is there somewhere that explains that concept a little clearer then wikipedia? also it looks like the numbers are a little off as i had made a type o on the interest rate of the 20 year.

[MDM]

is there somewhere that explains that concept a little clearer then wikipedia? also it looks like the numbers are a little off as i had made a type o on the interest rate of the 20 year.

Don't know if it will be clearer, but see below for a different way to look at it.  Won't change the answer of which is better, but different people see things better in different ways.

right now a 20 year mortgage will cost us 980 a month(3.635% interest) for P&I  30y is 798 (4.0%int)

The financed amount is $167,050 - close enough?

Assume you have $980.45/month.  You can
1) use that for 20 years to pay the 20 year mortgage (according to the Excel PMT function), then invest that amount each month for 10 years, or
2) use $797.52/mo for 30 years to pay the 30 year mortgage (again according to the Excel PMT function), while investing ($980.45 - $797.52 = $182.93) per month for those 30 years.

At the end of 30 years you would have
a) paid the mortgage, and
b) have some investment balance

Because the mortgage is paid either way, you just need to look at the investment balance at the end of the 30 years.

To get the ending value of a monthly investment after some number of months at some interest rate, you want the "Future value of an annuity" described in the wikipedia article.  This is also what Excel's FV function will give you.
In Excel,
FV(4%/12,10*12,-980.45) gives $144,371, and
FV(4%/12,30*12,-182.93) gives $126,962, so at a 4% market return the 20 year mortgage is better.

Using Jack's 7% assumption
FV(7%/12,10*12,-980.45) gives $169,701, and
FV(7%/12,30*12,-182.93) gives $223,168, so at a 7% market return the 30 year mortgage is better, as Jack found.

At 5% market return, the 20 and 30 year mortgages are ~equal.  Thus, as Jack said, YMMV.

You could copy the table below, paste into spreadsheet cell A1, then vary the inputs to your heart's content - good luck!

Loan amount	167050	=B1
Annual interest rate	0.03635	0.04
Loan period in years	20	30
Number of payments per year	12	=B4
Scheduled payment	=PMT(B2/B4,B3*B4,-B1)	=PMT(C2/C4,C3*C4,-C1)
Payment difference	=B5-C5
Number of years invested	=C3-B3	=C3
Market return	0.05	=B8
Investment balance at end	=FV(B8/12,B7*12,-B5)	=FV(C8/12,C7*12,-B6)
Advantage of the first choice	=B9-C9

[Jack]

is there somewhere that explains that concept a little clearer then wikipedia? also it looks like the numbers are a little off as i had made a type o on the interest rate of the 20 year.

Try this PDF, or search for "engineering economy." (There are certainly lessons on cash flow analysis outside of an engineering context, but I would expect something oriented towards accounting or business to be more detailed than you need.)

[Enigma]

I personally go for the lowest interest rate and pay extra principal to shrink the loan to 10 years.  First I save the almost 4% that would be going towards interest and whatever principal is paid off grows (albeit theoretically) as the price of the property goes up.

The house you live in is a large expense and one not needed when going through retirement eating up retirement funds.

[Jack]

I personally go for the lowest interest rate and pay extra principal to shrink the loan to 10 years.  First I save the almost 4% that would be going towards interest and whatever principal is paid off grows (albeit theoretically) as the price of the property goes up.

The house you live in is a large expense and one not needed when going through retirement eating up retirement funds.

I challenge you to do the math (as seen above) and post how much that choice is costing you.

[Enigma]

Lets look at some of the mistakes...

Option 2 (20-year mortgage)

PV_{option 2} = PV_mortgage + PV(FV_{investment at year 20})

Initial:
A_mortgage = -$980
i_mortgage = 3.635%/year = 0.00302916/month
n_mortgage = 240 months

PV_mortgage = (-$980/ 0.00302916) * (1 - (1/(1+0.00302916)²⁴⁰)) = -$166,973.12

From year 20 to year 30:
FV_{investment at year 20} = 0 (Invested nothing 20 years)
FV_{investment at year 20-30} = ($980/ 0.07) * (1 - (1/(1+0.07)¹²⁰)) = $84,403.83


-$139,794.09 < -$82,569.29, so the 20-year mortgage is better

[Enigma]

Also I have to look closer at the equations.  And review the website because saving $980/month for 10 years gives me =980x12x10 = $117,600

Also option 1 PVinvestment $182/month over 30 years ->$182x12x30 = $65,520

Again too many calculations that are semi-deceiving

That is me accepting the challenge to do the math

[Enigma]

Excel Loan Amortization
(4%, 30yr, 167k) - Total Interest Paid 35,718
(3.635%, 20yrs, 167k) - Total Interest Paid 27,830
*Principal is negated due to the fact you pay the same in both loans

-----------------------------------------------------------------------------------------------
Straight Savings Account
[30yr] $182/month over 30 years = $65,520
[20yr] $980/month for 10 years = $117,600
*Used as a baseline, somehow 0% interest was better than the equations thrown around

Using just a savings account 20 years is better
(30year) $29,800
(20year) $89,770
------------------------------------------------------------------------------------------------

However, compounding daily 7% over that time
(Starting=0, Rate=7%)
(Used an online calculator)
[30yr mortgage] $182/month over 30 years compounding at 7% = $224,207
[20yr mortgage] $980/month for 10 years compounding at 7%  = $170,801

[30yr] +188,489
[20yr] +142,971

*At 7% growth Average a 30 year is better

[MDM]

Using just a savings account 20 years is better
*At 7% growth Average a 30 year is better

Good to see that we agree.

[Jack]

Thank you, Enigma, for making me re-think my math. In doing so, I managed to confuse myself trying to figure out the right way to discount cash flows that don't start until some point in the future... anyway, long story short I mostly abandoned the formulas and made a spreadsheet with a full amortization table for each option instead. This should be a lot clearer!

TL;DR: 30-year still wins for any assumption of stock market returns 4.7% 5% or greater, and fully-amortizing a loan is better than paying the same loan off early unless the stock market return is less than the loan interest rate.

[MDM]

fully-amortizing [i.e., not paying off early] a loan is better than paying the same loan off early unless the stock market return is less than the loan interest rate.

Yes.

30-year still wins for any assumption of stock market returns 4.7% or greater

4.7% is essentially the same as 5% when we are talking about guessing Compound Annual Growth Rates over the next 30 years, so this is more from academic curiosity: the short spreadsheet table shown above gets 5% for the OP's choice - how did you arrive at 4.7%?

[Jack]

how did you arrive at 4.7%?

By making another mistake. Never mind.

[MDM]

Ok, thanks.  Already have the "math only" version of "pay mortgage vs. invest" in the 'Misc. calcs' tab of the case study spreadsheet.  Based on this thread it seems worthwhile to modify that slightly and have "mortgage option 1 vs. mortgage option 2, given an alternative market return."  It will look much like the example below when I get around to updating it.

Mortgage length vs. interest rate calculation.  Assumes the amount of Pmt2 is available. 
Compares investing the difference between Pmt2 and Pmt1 over the length of option 1 (n1) vs.
paying off the mortgage in fewer years (n2) and investing the full amount of Pmt2 for the next (n1 - n2) years.

Mortgage Principal	P	167,000	$
Mortgage length, option 1	n1	30	yr
Mortgage length, option 2	n2	20	yr
Mortgage interest rate, option 1	i1	4.0%
Mortgage interest rate, option 2	i2	3.635%
Investment interest rate	r	5.0%
Number of payments per year	Pmt/yr	12	/yr
Mortgage payment, option 1	Pmt1	797	$/mo
Mortgage payment, option 2	Pmt2	980	$/mo
Results from investing the difference in mortgage payments for n1 years	FV1	152,199	$
Results from investing the option 2 payment for (n1 - n2) years	FV2	152,201	$
Advantage of option 1	FV1 - FV2	-3	$

[crazysteve5575]

Thank you for all the numbers and explanation of the numbers.

looks like we are going to go with the 30 year mortgage.  Hope it works out.

[Fishindude]

I'm a firm believer in short term mortgages.
Saving a few bucks by going with a longer mortgage and investing that money instead sounds good, but truthfully most are not disciplined enough to contently do this.  the extra $$ will get blown on other things.   Your age is important too.  How old will you be in 30 years, and do you still want to be paying for this house at that age?   Taking the longer term mortgage and rationalizing that you have the option to pay it off early sounds good too, but most won't do this.  A short term mortgage forces you to pay it off in a timely fashion.

[Jack]

I'm a firm believer in short term mortgages.
Saving a few bucks by going with a longer mortgage and investing that money instead sounds good, but truthfully most are not disciplined enough to contently do this.  the extra $$ will get blown on other things.   Your age is important too.  How old will you be in 30 years, and do you still want to be paying for this house at that age?   Taking the longer term mortgage and rationalizing that you have the option to pay it off early sounds good too, but most won't do this.  A short term mortgage forces you to pay it off in a timely fashion.

Do we have to have this "math vs. psychology" debate in every thread? In this one, the OP clearly asked about the math, not the psychology.

[Dicey]

I'm a firm believer in short term mortgages.
Saving a few bucks by going with a longer mortgage and investing that money instead sounds good, but truthfully most are not disciplined enough to contently do this.  the extra $$ will get blown on other things.   Your age is important too.  How old will you be in 30 years, and do you still want to be paying for this house at that age?   Taking the longer term mortgage and rationalizing that you have the option to pay it off early sounds good too, but most won't do this.  A short term mortgage forces you to pay it off in a timely fashion.

Do we have to have this "math vs. psychology" debate in every thread? In this one, the OP clearly asked about the math, not the psychology.

IMO, until everyone understands the math and the psychology, responses such as fishindude's are reasonably appropriate. I don't happen to agree with him, but if it motivates others to examine their options, I support this kind of discussion. This shit really should be taught in school.

[K-ice]

I see you have taken the 30y option.

My advice was going to be it depends on what you do with the extra money.
As someone mentioned if you invest in stocks that gain 7% then the 30y is better.
If you just put the extra in a savings account you might as well pay it faster.


Option 1 $168K 20y @ 3.635% pmt $980 total payments $230,600
Option 2 $168K 30y @ 4.0%     pmt $799 total payment $287,600

Just because you have taken the 30y option doesn't mean you can't pay it faster.
For example: sign up for 30 pay it off like it's a 20y.

Option 3 $168K 20y @ 4.0%    pmt $1015 total payments $243,600

Of course this is not as good as the lower interest rate but it's not terrible.

We had a 25y mortgage but paid it off in 6 years. The interest paid was still $45,000 but we saved over $140,000 compared to the 25y plan.

For us it worked because we were not "investors" at the time. If you are not comfortable "investing" I would still try to pay it off faster than 30y.

[FiftyIsTheNewTwenty]

On top of Jack's analysis...

What about taxes, especially the capital gains exemption for primary residence, and whether your other investments are taxable?

[psyclotr0n]

Loan amount	167050	=B1
Annual interest rate	0.03635	0.04
Loan period in years	20	30
Number of payments per year	12	=B4
Scheduled payment	=PMT(B2/B4,B3*B4,-B1)	=PMT(C2/C4,C3*C4,-C1)
Payment difference	=B5-C5
Number of years invested	=C3-B3	=C3
Market return	0.05	=B8
Investment balance at end	=FV(B8/12,B7*12,-B5)	=FV(C8/12,C7*12,-B6)
Advantage of the first choice	=B9-C9

Question: the calc for Investment balance at end for the 20 year is based on the equivalent of investing that higher loan payment between years 20 and 30. Shouldn't it actually be from investing the lower amount over that period? Because that lower amount is the opportunity gain that would have otherwise been the payment between years 20-30...

[MDM]

Loan amount	167050	=B1
Annual interest rate	0.03635	0.04
Loan period in years	20	30
Number of payments per year	12	=B4
Scheduled payment	=PMT(B2/B4,B3*B4,-B1)	=PMT(C2/C4,C3*C4,-C1)
Payment difference	=B5-C5
Number of years invested	=C3-B3	=C3
Market return	0.05	=B8
Investment balance at end	=FV(B8/12,B7*12,-B5)	=FV(C8/12,C7*12,-B6)
Advantage of the first choice	=B9-C9

Question: the calc for Investment balance at end for the 20 year is based on the equivalent of investing that higher loan payment between years 20 and 30. Shouldn't it actually be from investing the lower amount over that period? Because that lower amount is the opportunity gain that would have otherwise been the payment between years 20-30...

Good question. 

The calculations assume the amount of Pmt2 is available each year, where Pmt2 is the higher annual payment corresponding to the shorter term.  Once the shorter term loan is paid, that full amount becomes available for investing.

With the longer term loan, Pmt1 is used for loan payments throughout that term while (Pmt2 - Pmt1) is available for investing each year.

Does that make sense?

[psyclotr0n]

With the longer term loan, Pmt1 is used for loan payments throughout that term while (Pmt2 - Pmt1) is available for investing each year.

Does that make sense?

Yes, it does, thanks. I've been trying to adopt it to a current scenario of whether to re-fi my current ~$500k to a 15-year at 2.875% vs keeping existing 30 year (with 26 years left) at 4%. The wrinkle, however, is that it's highly likely I'll sell in the next 3-5 years, so my thinking is that, if I sell at year 5, I'll have saved ~$30k in interest accrued, whereas the $800 /month extra payment, which is $48,000 over two years, and doing a quick and dirty estimate of an average balance of $24,000 (times 5% return = $1,200 of "income") it's clearly a huge advantage to re-fi. What am I missing, do you think?

[bigwhitedawg]

Same boat, Psyclotr0n.  It seems to me that what gets missed in this pure math calculation is how long are you holding the property and what do you believe the real estate market will do in the same time?  The latter, in some markets, could easily out perform even the 7% investment scenario.  Or am I off base?  New here myself... 

[MDM]

...what do you believe the real estate market will do in the same time?  The latter, in some markets, could easily out perform even the 7% investment scenario.

Yes, it could.  That doesn't mean it will.

It’s Difficult to Make Predictions, Especially About the Future.

[Linea_Norway]

I don't know how your mortgage will work, but when we had one, it was of a type where I could make extra payments when I wanted to. Any time when we had a good amount of excess money, I would make an extra payment on the mortgage and I asked the bank to let me pay the same amount as before and shorten the length of the mortgage. The alternative would have been to pay less to the bank every month. Eventually when you have the mortgage over a shorter time, you pay less rent in total.

[psyclotr0n]

I don't know how your mortgage will work, but when we had one, it was of a type where I could make extra payments when I wanted to. Any time when we had a good amount of excess money, I would make an extra payment on the mortgage and I asked the bank to let me pay the same amount as before and shorten the length of the mortgage. The alternative would have been to pay less to the bank every month. Eventually when you have the mortgage over a shorter time, you pay less rent in total.

Good insight, Linda. That's why I was looking into re-fi a 30 year to a 15 year; started thinking I'd pay down my existing mortgage, but if I was going to volunteer more, why not be rewarded with a lower interest rate I figured. But leaning towards not bothering just to save ~$300 in interest but have $800 less cash flow each month for longer than I'm likely to live here.