# What return and cashflow improvement do i get from pre-paying 10k on 2.3% 20 year mortgage?

#### SPC100

##### Frequent Poster
If you gave someone 10k at 2.3% for 20 years would you prefer to get it back as

230 euros per year for 20 years, and then the 10k back at the end (deposit a/c) OR
624 euros per year for 20 years, and 0 extra back at the end (mortgage prepayment)

Both are a 2.3% return, but they are two different series of cashflows.

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#### Brendan Burgess

##### Founder
Grand - would you mind editing your title and first post accordingly as I think you might mislead other people who might think that you have found a way of making magic money and shouldn't need to read 20 posts to find that you have not.

#### SPC100

##### Frequent Poster
I have edited title, and also edited the first post to answer the question to make it easier for future readers.

here is the proposed answer - in case folks would like to discuss more..

My financial return is 2.3%, but the series of cashflows that give me this return are different.

e.g. I think it is easier to understand this if you pretend you are the bank, e.g. imagine you were the bank and you loaned me 10k, I could pay it back it to you in two different ways:

230 euros per year for 20 years, and then the 10k back at the end (deposit account/interest only mortgage style)
624 euros per year for 20 years, and 0 extra back at the end (typical mortgage style - try Karl Jeacle mortgage calculator 10k loan, 20 years, 2.3%).

the cashflows you would get are very different, but the interest rate is still 2.3%

You can visualise/understand the higher cashflow as getting back some of your capital each year, instead of getting it all back at the end.

If you put your 10k in a hypothetical 2.3% after tax deposit a/c for 20 years, You get the former series of cashflows, if you pay off your mortgage you get the latter series of cashflows.

Nit: For me to end up with a similar sum from both cases at the end of 20 years, I would need to re-invest the additional cashflow at 2.3%

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##### Frequent Poster
Strictly speaking that's only true of an interest-only mortgage.

With an amortising mortgage, making a principal repayment ahead of schedule has a compounding effect because more of your subsequent (scheduled) payments go towards paying down principal (thereby further reducing the interest payments).
Sarenco I think the Boss is right - all interactions with your mortgage enjoy (suffer) the compound mortgage interest rate. The fact that compound interest over a number of years gives a higher figure when expressed as a simple interest is not of any economic relevance.
However, compounding does raise the interesting supplementary question - for how long do you enjoy the interest rate? The key to answering this question is to consider the Internal Rate of Return (IRR) of the change in cashflows. A few examples to explain.
Let's say I pay down an amount now but decide to keep the monthly repayments the same. Then the change in cashflow is a negative amount now with positive amounts at the back end as the mortgage gets paid off early. Let's say I pay just enough now to reduce my mortgage term from 30 years to 29 years then I enjoy an IRR of x% p.a. for 30 years.
If I pay enough to shorten the term by two years I enjoy x% p.a. for a mixture of 29 and 30 years and so on. If I pay off the whole mortgage I enjoy x% p.a. on a mixture from 1 year to 30 years, basically x% p.a. on the scheduled reducing balance.
In a sense OP's proposal is sub optimal as s/he is opting for a reduced repayment amount to keep the term the same. The savings are 2.3% p.a. but if the monthly repayment was maintained the 2.3% p.a. would be enjoyed on average for longer. In a sense reducing the monthly repayment is partly undoing the good work of paying an amount off early.
Similar considerations apply in the interest only scenario. For whilst you gain x% p.a. on the amount paid off, you enjoy that in the compound sense for a lesser duration because you reduce the regular (interest) payments.

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#### Brendan Burgess

##### Founder
Duke, while I am delighted to be confirmed right, I wonder if we are making it a bit too complicated?

SPC can get a return on his money of 2.3% net by paying a lump sum off his mortgage.

He is not getting the 6% he originally thought he was getting, which is the main thing.

I don't think he should be planning for 10 years, much less 29 years.

He is getting that return at the moment and as it's the best return, he should avail of it.

If after 5 years, the mortgage rate has dropped to 1% and he can get 5% net by investing in savings bonds, then he should decide at that stage what to do with any further capital.

Brendan

##### Frequent Poster
Boss I think it is that complicated. OP has stated that s/he will pay off 10k but leave the term the same. The change in cash flow between her/him and the mortgage provider is +10k now followed by -624 p.a. for 20 years. Yes s/he will enjoy a saving of 2.3% p.a. on the 10k but she will suffer a loss of 2.3% p.a. on the 624 p.a.
By keeping the term the same s/he is partly undoing the good work of paying off the 10k. S/he still gets an overall compound return of 2.3% p.a. on the 10k but for a lesser duration on average.

#### Brendan Burgess

##### Founder
S/he still gets an overall compound return of 2.3% p.a. on the 10k but for a lesser duration on average.
I agree with that calculation but it is very unlikely that she will have the mortgage in 20 years. She may have paid off the mortgage or she may trade up or whatever.

I like planning for the next two or three years to put a solid foundation in place for the rest of one's life. But I hate when people say "if you pay off €10k now, you will save €12k in interest over 21 years".

Brendan

##### Frequent Poster
My point is a bit more than pedantic. Think of the greedy building society manager. Here is her reaction. "Damn, they're paying down 10k! But at least they are reducing their repayments by 640 p.a. That goes some way to make up for it." So I think the stock AAM advice should be that if you have a few bob to spare, pay down your mortgage but keep up the level of repayments you were used to, don't be fooled into accepting a reduced monthly repayment amount, that's what the B/S wants.

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#### dub_nerd

##### Frequent Poster
EDIT: Dang! Every time I go to post a bit of maths, Duke of Marmalade is there before me . Anyway, just to confirm what he and de boss said ...

Paying a lump sum off your mortgage has the same effect as a compound interest investment on the lump sum as long as you shorten the mortgage term (rather than reducing the amount of your repayments and keeping the same term -- that's a whole different calculation).

Have a look at the mortgage formulas on Wikipedia, keeping the same nomenclature:
• P = principal
• r = interest rate
• N = number of periods
• c = mortgage repayment per period
Look at the Total Interest Paid formula after N periods, assuming interest rate and periodic payment are constant. The total interest paid is therefore a function of Principal and Number of periods:

$I(P,N)=(Pr-c)\frac{(1+r)^N-1}{r}+cN$

The difference in interest paid if we pay off lump sum L up front is:

$I(P,N)-I(P-L,N)=(P-(P-L))r)\frac{(1+r)^N-1}{r}=L((1+r)^N-1)$

The right hand side is exactly the same as the compound interest formula for a lump sum L. Note that N will be the reduced term.

#### Sarenco

##### Frequent Poster
Paying a lump sum off your mortgage has the same effect as a compound interest investment on the lump sum as long as you shorten the mortgage term (rather than reducing the amount of your repayments and keeping the same term -- that's a whole different calculation).
Yes, that's the point I was trying to make but no doubt expressed poorly.

##### Frequent Poster
sorry for stealing your thunder dub_nerd. I think your formula needs a slight correction. If we set N equal 0, i.e. pay the mortgage off now, we get a saving of 0, which is obviously not correct. The correct formula (I think) is:
I(P,Nbefore) - I(P-L,Nafter)

#### Brendan Burgess

##### Founder
So I think the stock AAM advice should be that if you have a few bob to spare, pay down your mortgage but keep up the level of repayments you were used to,
Almost right.

The correct advice is to retain the mortgage term so that your scheduled repayments are lower.

But then make the full repayments anyway.

That way if you run into a problem later, you can return to making the lower repayments without asking the lender to reschedule you.

Brendan

#### SPC100

##### Frequent Poster
Boss I think it is that complicated. OP has stated that s/he will pay off 10k but leave the term the same. The change in cash flow between her/him and the mortgage provider is +10k now followed by -624 p.a. for 20 years. Yes s/he will enjoy a saving of 2.3% p.a. on the 10k but she will suffer a loss of 2.3% p.a. on the 624 p.a.
By keeping the term the same s/he is partly undoing the good work of paying off the 10k. S/he still gets an overall compound return of 2.3% p.a. on the 10k but for a lesser duration on average.
EDIT: Dang! Every time I go to post a bit of maths, Duke of Marmalade is there before me . Anyway, just to confirm what he and de boss said ...

Paying a lump sum off your mortgage has the same effect as a compound interest investment on the lump sum as long as you shorten the mortgage term (rather than reducing the amount of your repayments and keeping the same term -- that's a whole different calculation).
Technically ye are both late ;-) My nit in post 23, and earlier in the thread already addressed this - to get a compounded 2.3% on my 10k, I would need to be able re-invest each cashflow saving at 2.3%.

Thanks for highlighting that the simple way to do that, is to shorten the scheduled loan duration.

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#### SPC100

##### Frequent Poster
FWIW, in my case, I specifically did not want to shorten my term. I wanted additional cash flow. I know I will invest and not 'lifestyle spend' the additional cash flow.

I wanted to de-risk a bit. I had a very concentrated exposure, that had grown to be a significant portion of my net worth (not bitcoin!). My mortgage balance was larger than I was comfortable with. So, In Aug after selling 50% of the position, I was considering either re-investing the money in a (set of) broad stock market indices or paying down my mortgage or looking to buy a property. (Pension already maxed).

While I'm generally a long term buy and hold investor, and am aware of the studies that show lump sum stockmarket investment gives better return than drip-feeding in vast majority of cases. And I try not to hold more cash than an emergency fund. I would not have been able to sleep soundly putting it all in broad indexes in Aug.

I compromised by paying down the mortgage significantly in Aug to reduce my monthly outgoings, and planned to use the additional future monthly cashflow to drip-feed into the market, Although I haven't managed to set the drip feed up yet - I hope to sort that out over the next few weeks.

I was surprised by how significant, the cash flow increase was, and hence this thread, for me to understand why the cashflow increase was closer 6% than 2%.

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##### Frequent Poster
FWIW, in my case, I specifically did not want to shorten my term. I wanted additional cash flow. I know I will invest and not 'lifestyle spend' the additional cash flow.

I wanted to de-risk a bit. I had a very concentrated exposure, that had grown to be a significant portion of my net worth (not bitcoin!). My mortgage balance was larger than I was comfortable with. So, In Aug after selling 50% of the position, I was considering either re-investing the money in a (set of) broad stock market indices or paying down my mortgage or looking to buy a property. (Pension already maxed).

While I'm generally a long term buy and hold investor, and am aware of the studies that show lump sum stockmarket investment gives better return than drip-feeding in vast majority of cases. And I try not to hold more cash than an emergency fund. I would not have been able to sleep soundly putting it all in broad indexes in Aug.

I compromised by paying down the mortgage significantly in Aug to reduce my monthly outgoings, and planned to use the additional future monthly cashflow to drip-feed into the market, Although I haven't managed to set the drip feed up yet - I hope to sort that out over the next few weeks.

I was surprised by how significant, the cash flow increase was, and hence this thread, for me to understand why the cashflow increase was closer 6% than 2%.
Good stuff SPC You have obviously thought out your strategy well. Channeling savings on mortgage repayments into equities is of course a form of geared investment strategy. If your horizon is sufficently long then 2.3% could be viewed as a reasonable price to pay to tap into the Equity Risk Premium which over the long term should cover those borrowing costs.

#### SPC100

##### Frequent Poster
Yes, I do agree, and furthermore, anyone who holds a mortgage/loan and chooses to invest more or hold on to their existing investments rather than pay down the mortgage, is effectively borrowing at their mortgage rate to fund their investment.

Duke, that seems very carefully worded, but it sounds like you think it has long term expected value. Although my mortgage costs are not fixed at 2.3% for twenty years, and we don't know what the future return will be.

If the gods of longevity smile on me, I would hope to be personally holding some of these equity purchases (or their logical descendants) for periods of between 10-50 years.

#### dub_nerd

##### Frequent Poster
sorry for stealing your thunder dub_nerd. I think your formula needs a slight correction. If we set N equal 0, i.e. pay the mortgage off now, we get a saving of 0, which is obviously not correct. The correct formula (I think) is:
I(P,Nbefore) - I(P-L,Nafter)
Just had to spoil it, didn't ya. I see your point. Though my version is nice and simple and shows the saving due to the lump sum after any N periods, and it conveniently corresponds to the compound interest on the lump sum over that period. Arguably after 0 periods you haven't saved anything yet. But yes, of course, when the lump sum version of the mortgage is finished the non-lump sum version still has more time to run which I haven't accounted for. The problem is it's not trivial to work out how many periods that is. However, here goes:

The formula for the amount remaining after N constant payments of c is:

$(1+r)^NP-\frac{(1+r)^N-1}{r}c$

Normally N is a fixed term and we set the remaining amount to zero and solve for c:

$c=\frac{r}{1-(1+r)^{-N}}P$

But if we then take this c and hold it constant we can solve the first equation instead for "N after"

$N_a=\frac{\log\left(\frac{c}{c-rP} \right )}{\log(1+r)}$

Then, to get the total interest savings over the full life of the mortgage with or without lump sum L, we have the very unwieldly:

$\Delta I=c(N-N_a)-L=\frac{rP}{1-(1+r)^{-N}}\left (N-\frac{\log\left(\frac{c}{c-r(P-L)} \right )}{\log(1+r)}\right )-L$

Amazingly it actually works. For the OP's example of €100k over 20 years at 2.3%, with optional lump of €10k it gives:

$c=\euro 520.21,\ N_a=210.4\text{ (months)}, \Delta I=\euro 5403.87$

This ups the notional rate of return on the lump sum to 2.47%. It seems odd to me that it's higher than the mortgage rate, but I guess that's the bonus for paying it off early.

##### Frequent Poster
$c=\euro 520.21,\ N_a=210.4\text{ (months)}, \Delta I=\euro 5403.87$

This ups the notional rate of return on the lump sum to 2.47%. It seems odd to me that it's higher than the mortgage rate, but I guess that's the bonus for paying it off early.

I return to my earlier comment. I think (but not sure) that you have done this calculation (2.47%) on the basis that L earns the interest over a period of Na years. But in fact it earns interest on all of L over Na years and on a reducing balance between Na and N years. This can be seen most clearly when Na is 0. The answer will always be the mortgage interest rate (I think).

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#### dub_nerd

##### Frequent Poster
Not getting you there. You seem to be suggesting the lump saves/earns money over longer than the life of the mortgage. But if you pay the lump sum the mortgage only lasts Na periods (even though we're calculating the interest saved compared to the full mortgage of N periods). The notional interest rate is just calculated from the avoided interest, spread over Na periods. Specifically:

$r=\left ( 1+\frac{\Delta I}{L}\right )^{1/N_a}-1$

(The only liberty I took was monthly compounding, as an AER it should probably be 2.49%).

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##### Frequent Poster
Not getting you there. You seem to be suggesting the lump saves/earns money over longer than the life of the mortgage. But if you pay the lump sum the mortgage only lasts Na periods (even though we're calculating the interest saved compared to the full mortgage of N periods). The notional interest rate is just calculated from the avoided interest, spread over Na periods. Specifically:

$r=\left ( 1+\frac{\Delta I}{L}\right )^{1/N_a}-1$

(The only liberty I took was monthly compounding, as an AER it should probably be 2.49%).
Sorry, this calculation obviously does not work if Na is 0 i.e. if the full mortgage is paid off. Or if you prefer it obviously gives a silly answer if L is big enough to reduce Na to 1.

The return on L is by reference to the change in the original cashflow which is over N years. The IRR gives the return over N years albeit it is only enjoyed compoundly on the whole L for Na years and then on a reducing balance from Na to N years.

All changes to the cashflow on a mortgage are at an IRR equal to the mortgage rate. All that is at issue is for how much and for how long does the IRR compound.