## Can someone check my math and my logic on this?

Non-investing personal finance issues including insurance, credit, real estate, taxes, employment and legal issues such as trusts and wills
Topic Author
Bushka
Posts: 14
Joined: Sat Nov 14, 2020 1:13 pm

### Can someone check my math and my logic on this?

Hi. First post here, hope it isn't a bonehead one!

Without getting into too much detail at the moment, I want to make a comparison and I'm using hypothetical numbers. Let's say I have \$100K in cash, and I also have a new debt of \$100K that is to be paid over 10 years at 5% interest, which is a monthly payment of \$1060. I'm trying to decide whether to pay the entire \$100K up front and do away with the debt, or to take on the debt and invest the \$100K. This is the way I was looking at it:

Option 1: Pay off the loan now and take the monthly payment I would have had (\$1060) and invest it.

Option 2: Take on the debt and invest the \$100K now, incurring the \$1060 monthly payment.

How to compare? I made the assumption that an investment over 10 years would work out to 5% compounded interest through the stock market and other diversification. I chose 5% because it is the same as the interest on the debt and I just wanted to kind of compare apples to apples. The market typically would outperform 5% interest over 10 years or so (I know, no guarantee of that). I ran compound interest calculations that showed the \$100K lump sum would be worth about \$164K in 10 years. Interestingly, if I paid off the debt and assume that I make \$1060 monthly investments starting from 0, I also end up at about \$164K in 10 years (\$1060/mo is about \$127K in 10 years).

So I figure either way I'm going to end up with \$164K in investment income (theoretically). With Option 1 I lose \$100K for a net of \$64K. With Option 2 I lose the principal AND interest to the creditor, which is \$127K. So in that case my net is 164 - 127 = \$37K. That would suggest that I should pay off the debt right away and hope I can beat the interest rate in the market or elsewhere.

I know there are other issues to consider, but mathematically am I looking at this in a reasonable way?
RubyTuesday
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Joined: Fri Oct 19, 2012 11:24 am

### Re: Can someone check my math and my logic on this?

The wiki has a useful article. Paying Down Loans vs Investing

Bottom line for me is that paying the loan off is basically 5% guaranteed return. Can’t beat that. If you have sufficient emergency funds, discipline to invest the payment, etc, pay the loan and invest the cash flow.
“Doing nothing is better than being busy doing nothing.” – Lao Tzu
delamer
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Joined: Tue Feb 08, 2011 6:13 pm

### Re: Can someone check my math and my logic on this?

You should be taking taxes into account. How much do you need to earn to net \$1060/month after tax? How much taxes will you pay on the stock earnings?

(These will vary depending on the source of income and where funds are deposited.)

BTW, under your original assumptions of the loan rate and investment return being the same amount (5% in your example), then the math should work out to you ending up with the same amount of money after 10 years. If it didn’t, that would mean that your math was wrong.
Last edited by delamer on Mon Nov 16, 2020 6:29 pm, edited 1 time in total.
Pacific
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Location: Lost in the middle of the Pacific

### Re: Can someone check my math and my logic on this?

My vote is for Option 1.

I did this and never looked back.

I hate being in debt for anything more than my monthly credit card bill.
retire2022
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Location: NYC

### Re: Can someone check my math and my logic on this?

op

There was this posting from a few days ago which walk through the thinking process, which could apply to your situation in case you like to read it.

viewtopic.php?f=2&t=329229
FiveK
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### Re: Can someone check my math and my logic on this?

Bushka wrote: Mon Nov 16, 2020 5:37 pm I know there are other issues to consider, but mathematically am I looking at this in a reasonable way?
Yes. If the after-tax return on the investments is identical to the after-tax interest rate on the loan, the results will be identical no matter when you pay off the loan. Well done!
FiveK
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Joined: Sun Mar 16, 2014 2:43 pm

### Re: Can someone check my math and my logic on this?

Bushka wrote: Mon Nov 16, 2020 5:37 pm So I figure either way I'm going to end up with \$164K in investment income (theoretically). With Option 1 I lose \$100K for a net of \$64K. With Option 2 I lose the principal AND interest to the creditor, which is \$127K. So in that case my net is 164 - 127 = \$37K. That would suggest that I should pay off the debt right away and hope I can beat the interest rate in the market or elsewhere.
But if you think your after-tax investment return will be greater than your after-tax loan interest, paying the minimum on the loan while continuing to invest is the logical conclusion.

Your spreadsheet (or whatever calculations) should show this. If not, check for errors....

It often remains a trade-off between a lower guaranteed return vs. a higher non-guaranteed return.

On a psychological basis, some people hate debt and others hate giving up the opportunity for higher returns. Neither group is inherently correct, nor do they often understand how the other group could possibly think that way.
RyeBourbon
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Location: NJ/Philly

### Re: Can someone check my math and my logic on this?

5% is pretty high in this rate environment. I would pay that off as fast as possible.
joylesshusband
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Joined: Thu Jan 18, 2018 6:35 pm

### Re: Can someone check my math and my logic on this?

Bushka wrote: Mon Nov 16, 2020 5:37 pm am I looking at this in a reasonable way?
No, you are not.
What you missed is that the investment and the debt have 2 very different timelines.

With Option 1 you retire the debt right away, but your \$100k are gone forever. They will never earn you a single dollar.
With Option 2 your \$100k investment does not have just a 10 year time horizon as you assumed. Instead, its time horizon is until your death, meaning that it will keep earning much longer after the debt is paid off.
Hence, the differential between the 2 options would be a lot more than the \$37 k you surmised.

Debt has a fixed term, while investments have term equal to investor's lifetime.
Retired July 2018 @ age 59. Posting here purely for amusement.
FiveK
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Joined: Sun Mar 16, 2014 2:43 pm

### Re: Can someone check my math and my logic on this?

joylesshusband wrote: Mon Nov 16, 2020 9:00 pm
Bushka wrote: Mon Nov 16, 2020 5:37 pm am I looking at this in a reasonable way?
No, you are not.
What you missed is that the investment and the debt have 2 very different timelines.

With Option 1 you retire the debt right away, but your \$100k are gone forever. They will never earn you a single dollar.
With Option 2 your \$100k investment does not have just a 10 year time horizon as you assumed. Instead, its time horizon is until your death, meaning that it will keep earning much longer after the debt is paid off.
Hence, the differential between the 2 options would be a lot more than the \$37 k you surmised.

Debt has a fixed term, while investments have term equal to investor's lifetime.
Compare the situation at the end of the nominal loan length, and the investor's lifetime (assuming it's greater than the loan length) doesn't matter.

However fast one pre-pays the loan, ranging from immediately to no extra payments, at the end of the nominal loan length the person has
a) a paid-off loan, and
b) some invested amount "X"

Whatever approach leads to the largest value of "X" will have been mathematically best. We just don't know ahead of time what approach that is, leading to the never-ending debate....
venkman
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Joined: Tue Mar 14, 2017 10:33 pm

### Re: Can someone check my math and my logic on this?

Bushka wrote: Mon Nov 16, 2020 5:37 pm Without getting into too much detail at the moment, I want to make a comparison and I'm using hypothetical numbers. Let's say I have \$100K in cash, and I also have a new debt of \$100K that is to be paid over 10 years at 5% interest, which is a monthly payment of \$1060. I'm trying to decide whether to pay the entire \$100K up front and do away with the debt, or to take on the debt and invest the \$100K. This is the way I was looking at it:

Option 1: Pay off the loan now and take the monthly payment I would have had (\$1060) and invest it.

Option 2: Take on the debt and invest the \$100K now, incurring the \$1060 monthly payment.
With Option 2, you would effectively be borrowing money at 5% interest to invest in (presumably) stocks. You would be taking 100% of the equity risk to get 500 basis points less than the market return.

You might come out ahead on an absolute basis. You are almost guaranteed to lose on a risk-adjusted basis.
inbox788
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### Re: Can someone check my math and my logic on this?

delamer wrote: Mon Nov 16, 2020 6:13 pm You should be taking taxes into account. How much do you need to earn to net \$1060/month after tax? How much taxes will you pay on the stock earnings?

(These will vary depending on the source of income and where funds are deposited.)

BTW, under your original assumptions of the loan rate and investment return being the same amount (5% in your example), then the math should work out to you ending up with the same amount of money after 10 years. If it didn’t, that would mean that your math was wrong.
OP, ignoring taxes and other more important considerations and only looking at your comparison, the math is wrong. You forgot compound growth on your investment! If you add that in, you would be square.
Topic Author
Bushka
Posts: 14
Joined: Sat Nov 14, 2020 1:13 pm

### Re: Can someone check my math and my logic on this?

Thanks for all the replies! I do not seem to be getting email notifications of replies so I was unaware of the activity to date. I do have some follow ups when I get a chance to go through everything.

Dan
Shallowpockets
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### Re: Can someone check my math and my logic on this?

Your next question on BHs will be how to invest that \$100k and you will be flummoxed by that. You could be standing still, wondering, and still paying that \$1060 a month.

Or, you will not have a next question due to paying the loan off. You will sleep better and we will sleep better.
MBB_Boy
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Joined: Sat May 12, 2018 4:09 pm

### Re: Can someone check my math and my logic on this?

Something seems off. 100K invested right now should come out to more than 1060/month over 5 years

Edit: Nvm, had the number of years wrong

To OP, this is a risk tolerance decision. On here and other financial forums, it is "generally agreed" that investing comes ahead of low interest debt (1-3%), and debt payoff comes first for high interest debt (7%+). You've discovered the range (4-6%) in which there can be no consensus, and personal finance becomes incredibly personal.

If it's me, I lean toward investing. Other factors to consider would include the size of your current investments, how high your savings rate is, and how stable you feel in employment. The danger (to me) of having the debt and required payment comes if you find yourself in a situation where your income is impacted. If the payment does not stretch you thin and you have ample savings / investments, that tilts the scale toward taking the risk that the market beats 5% in the long term (history says yes). If having the payment stresses you out and you don't have extra income to pay it AND save to get ahead, maybe you get rid of the debt to derisk your day-to-day.

And of course, do both if you're indecisive! Invest some of the cash, use the rest to pay down the debt.

(Although you're tactically better off holding that debt payoff money in a savings account somewhere and working to pay the loan off early out of income, because there's little benefit to paying the debt down until you pay it off completely. You still have to make the payment until its gone, and you'd give up liquidity)
Topic Author
Bushka
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Joined: Sat Nov 14, 2020 1:13 pm

### Re: Can someone check my math and my logic on this?

delamer wrote: Mon Nov 16, 2020 6:13 pm BTW, under your original assumptions of the loan rate and investment return being the same amount (5% in your example), then the math should work out to you ending up with the same amount of money after 10 years. If it didn’t, that would mean that your math was wrong.
That's not quite what I did. I think you are saying that owing 100K for 10 years at 5% is the same as investing 100K for 10/5%. Then the total would be the same. I figured that if I paid off the 100K now instead of paying \$1060 each month, I could take that same \$1060 and invest it instead. So I'm really taking the principal and interest from the debt and then investing that for a 5% assumed gain. That's why that option generated more money.
jerrysmith
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### Re: Can someone check my math and my logic on this?

Option 1 all day.
FiveK
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### Re: Can someone check my math and my logic on this?

Bushka wrote: Mon Nov 16, 2020 5:37 pmSo I figure either way I'm going to end up with \$164K....
Ah, I thought you meant you'll have the same investment balance whether you pay the loan minimum, all of it up front, or anything in between. For identical loan interest and investment return rates, that is indeed what happens.
With Option 2 I lose the principal AND interest to the creditor, which is \$127K. So in that case my net is 164 - 127 = \$37K. That would suggest that I should pay off the debt right away and hope I can beat the interest rate in the market or elsewhere.
Don't worry about how much you are paying to someone else, whether that is a creditor, the IRS, etc. Instead, look at how much you have after all taxes are paid. See the second of two Common misconceptions for one example where paying more tax in absolute terms could still leave you with more money after tax.
I know there are other issues to consider, but mathematically am I looking at this in a reasonable way?
Take that \$100K loan amount for 10 years at 5%, except assume an annual payment that occurs at the beginning of the year. That payment would be \$12,333.77. Also assume a cash stream of \$100K/yr that arrives on the first of every year.

One can choose to direct any part of that \$100K/yr to the loan, from the full amount remaining on the loan principal, down to \$12,333.77. Whatever is not used to pay the loan goes to investments that also return 5%/yr (and both 5% numbers are after tax).

You should find that at the end of year 10, the loan is paid and the investment account has \$1,157,789.25 - regardless of how fast the loan was paid.

Does that make sense?
Topic Author
Bushka
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Joined: Sat Nov 14, 2020 1:13 pm

### Re: Can someone check my math and my logic on this?

It makes sense, but that isn't really my situation. Wish it were! I don't have \$100K/year disposable income to invest.
FiveK
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### Re: Can someone check my math and my logic on this?

Bushka wrote: Sat Nov 21, 2020 1:42 pm It makes sense, but that isn't really my situation. Wish it were! I don't have \$100K/year disposable income to invest.
Pick any number, but the point is the same: investing, or paying a loan, with the same return/interest, gives identical results.
Topic Author
Bushka
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Joined: Sat Nov 14, 2020 1:13 pm

### Re: Can someone check my math and my logic on this?

FiveK wrote: Sat Nov 21, 2020 2:15 pm
Bushka wrote: Sat Nov 21, 2020 1:42 pm It makes sense, but that isn't really my situation. Wish it were! I don't have \$100K/year disposable income to invest.
Pick any number, but the point is the same: investing, or paying a loan, with the same return/interest, gives identical results.
OK, I wanted to be sure you didn't think I was investing 100K each year. On a slightly different note, if I have \$100K to invest and I make no more contributions I will have accumulated more interest than if I started with \$0 and invested about \$833/mo for 10 years (equals \$100K). It's about \$165K vs \$125K. In my example I chose to invest \$1060 instead of \$833 because I figure if I have \$1060 to pay the debtor then I have \$1060 to invest. If I do that I also end up (coincidentally?) at about \$165K. I'm not sure of what point I'm making other than rolling these numbers around to make sure I understand.

Also, thank you to everybody who replied. I've read all the links and have a little better perspective on this.
howard71
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### Re: Can someone check my math and my logic on this?

I think there are just too many unknown variables involved to reduce this problem to simple math and logic.

You don't know what kind of return you will get on your investment and you don't know if your house is going to increase or decrease in value over time and how long you will keep it. You can only make your best educated guesses at those things.

So I think it all boils down in the end to what floats your boat. I would opt for the mortgage but that's just me.
FiveK
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### Re: Can someone check my math and my logic on this?

Bushka wrote: Sat Nov 21, 2020 3:34 pm ...if I have \$100K to invest and I make no more contributions I will have accumulated more interest than if I started with \$0 and invested about \$833/mo for 10 years (equals \$100K). It's about \$165K vs \$125K.
Yes - definitely better to have \$100K in hand now, instead of receiving a total of \$100K in monthly payments over the next 10 years. See Time value of money.
In my example I chose to invest \$1060 instead of \$833 because I figure if I have \$1060 to pay the debtor then I have \$1060 to invest.
Yes, and if the investment return rate equals the debt interest rate, you end up with exactly the same investment balance at the end of the nominal debt repayment term, no matter how much you pay above the minimum on the debt.
BogleFan510
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### Re: Can someone check my math and my logic on this?

Its better to think of the stock side as a range of possible outcomes, rather than one outcome, assuming a 'typical' compound rate of return. Stocks dont work like that.

While your math is a good starting point, the truth is that you may come out ahead, even, or behind assuming a stock investment. There is a distribution of possible outcomes, one that is currently unknown. In theory you could lose 50% or more of the stock investment and still owe the money (worst case, low probability).
Topic Author
Bushka
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### Re: Can someone check my math and my logic on this?

@FiveK: I've seen that ring nebula in my telescope many years ago. Amazing thing, although of course it only looks like a white smudge in real life. The ring is clearly distinct, though.
Topic Author
Bushka
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Joined: Sat Nov 14, 2020 1:13 pm

### Re: Can someone check my math and my logic on this?

BogleFan510 wrote: Sat Nov 21, 2020 4:15 pm Its better to think of the stock side as a range of possible outcomes, rather than one outcome, assuming a 'typical' compound rate of return. Stocks dont work like that.

While your math is a good starting point, the truth is that you may come out ahead, even, or behind assuming a stock investment. There is a distribution of possible outcomes, one that is currently unknown. In theory you could lose 50% or more of the stock investment and still owe the money (worst case, low probability).
Yes, good point. I do have a real life situation coming up, which is why I joined this forum. Great feedback so far. I am going to post the actual situation in another thread as soon as I put all my thoughts together.
FiveK
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### Re: Can someone check my math and my logic on this?

Bushka wrote: Sat Nov 21, 2020 4:21 pm @FiveK: I've seen that ring nebula in my telescope many years ago. Amazing thing, although of course it only looks like a white smudge in real life. The ring is clearly distinct, though.
Yes, the Hubble camera has its advantages.
Let's say I have \$100K in cash, and I also have a new debt of \$100K that is to be paid over 10 years at 5% interest...I'm trying to decide whether to pay the entire \$100K up front and do away with the debt, or to take on the debt and invest the \$100K.
...
...assumption that an investment over 10 years would work out to 5%...
...
I know there are other issues to consider, but mathematically am I looking at this in a reasonable way?
Another quick way to look at this - you could:
- Pay the debt immediately. You then have no debt and no cash.
- Invest the \$100K at 5%/yr return. After the first month, withdraw the \$1060.66 needed for the debt payment. You then have \$99,356.01 in the investment account, and a remaining balance of \$99,356.01 on your debt.

At this point you could:
- Pay the debt immediately. You then have no debt and no cash.
- Allow the investment to grow for another month. After that month, withdraw the \$1060.66 needed for the debt payment. You then have \$98,709.34 in the investment account, and a remaining balance of \$98,709.34 on your debt.

Repeat until the debt is paid and you have no investment, which will happen at the same time, whatever time that is.

And yes, that's the pure math look, while in the real world the investment is highly unlikely to behave that way. Might be better, might be worse....
Topic Author
Bushka
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Joined: Sat Nov 14, 2020 1:13 pm

### Re: Can someone check my math and my logic on this?

FiveK wrote: Sat Nov 21, 2020 4:47 pm
Bushka wrote: Sat Nov 21, 2020 4:21 pm @FiveK: I've seen that ring nebula in my telescope many years ago. Amazing thing, although of course it only looks like a white smudge in real life. The ring is clearly distinct, though.
Yes, the Hubble camera has its advantages.
Let's say I have \$100K in cash, and I also have a new debt of \$100K that is to be paid over 10 years at 5% interest...I'm trying to decide whether to pay the entire \$100K up front and do away with the debt, or to take on the debt and invest the \$100K.
...
...assumption that an investment over 10 years would work out to 5%...
...
I know there are other issues to consider, but mathematically am I looking at this in a reasonable way?
Another quick way to look at this - you could:
- Pay the debt immediately. You then have no debt and no cash.
- Invest the \$100K at 5%/yr return. After the first month, withdraw the \$1060.66 needed for the debt payment. You then have \$99,356.01 in the investment account, and a remaining balance of \$99,356.01 on your debt.

At this point you could:
- Pay the debt immediately. You then have no debt and no cash.
- Allow the investment to grow for another month. After that month, withdraw the \$1060.66 needed for the debt payment. You then have \$98,709.34 in the investment account, and a remaining balance of \$98,709.34 on your debt.

Repeat until the debt is paid and you have no investment, which will happen at the same time, whatever time that is.

And yes, that's the pure math look, while in the real world the investment is highly unlikely to behave that way. Might be better, might be worse....
Maybe you know this, but Hubble does not technically take color photos. Some of the images are close to what you might see if you went there in a spaceship, but many are color enhanced to bring out more detail, much of which is not visible to the naked eye.

Thanks for the idea about kind of doing both at the same time. Interesting thought.
FiveK
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### Re: Can someone check my math and my logic on this?

Bushka wrote: Sun Nov 22, 2020 6:07 am
FiveK wrote: Sat Nov 21, 2020 4:47 pm Yes, the Hubble camera has its advantages.
Maybe you know this, but Hubble does not technically take color photos. Some of the images are close to what you might see if you went there in a spaceship, but many are color enhanced to bring out more detail, much of which is not visible to the naked eye.
I was thinking more of "being above the atmosphere" and (probably) "larger mirror" but that's a good point: Truth Behind the Photos: What the Hubble Space Telescope Really Sees | Space
Thanks for the idea about kind of doing both at the same time. Interesting thought.
Perhaps even simpler: the equations for loan and investment balance, using one compounding period and subsequent payment/withdrawal:

Loan ending balance = (Loan beginning balance) * (1 + interest_rate) - payment
Investment ending balance = (Investment beginning balance) * (1 + return_rate) - withdrawal

When the respective terms on the right are equal, the ending balances are equal.