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



## SPC100 (17 Dec 2018)

Hi,

I always considered the return from reducing my mortgage was the same as the interest rate on my mortgage i.e. 2.3%. But I never paid much attention to the cashflow benefit (if you leave the term the same).

The calculations below, indicate after paying 10k on a 20 year 2.3% Mortgage, I get 624 Euro / 6.2% back per year.

How do I explain this illusory return?
*
Calculations (from drjeacle calculator)*
100k mortgage, 20 years, 2.3% = 520.21 per month repayment
90k mortgage, 20 years, 2.3% = 468.18 per month repayment

If I 'invest' 10k in mortgage repayments, I gain 52.03 per month (520.21-468.18), 624.36 per year.

624.36*100/10,000 = 6.2436% Return p.a.

_

==============================
I edited this post to add the answer here - As this turned into a long thread, and brendan was concerned people might get mislead_
*

The answer:*

*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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## RedOnion (17 Dec 2018)

You're including your capital as part of your return.

The net is your interest rate (2.3%).


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## SPC100 (17 Dec 2018)

I used to always count it at 2.3%, but after a recent large mortgage payment, I noticed the cashflow gain was much higher.

I see how if it was an interest only loan, my gain would be only 2.3%, and the calcs confirm it:

100k mortgage, interest only, 2.3% = 191.67 per month repayment
90k mortgage, interest only, 2.3% = 172.50 per month repayment
(191.67-172.50)*12*100/10,000)=2.3%

And If I do the calcs for e.g. a 100 year mortgage, the 'gain' is calculated to be 2.6% of my 10k p.a. (over 100 years the annual capital payment is very small).

I don't think i have a good internal visualisation/representation of how my capital is included in the return. Can you explain it a bit more please?


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## RedOnion (17 Dec 2018)

You're counting the entire reduced outward cashflow as return, but not factoring in the initial large negative cashflow (repayment) as far as I can see?


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## Zenith63 (17 Dec 2018)

Seems to me you've 'invested' €10000 which will allow you pay out €12485.13 less over the course of the mortgage, so a €2485.13 return/'profit' over 20 years.  Or 1.24% per annum?  Mortgage calculations are so unintuitive...


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## Brendan Burgess (17 Dec 2018)

Zenith63 said:


> Mortgage calculations are so unintuitive...



No, they are not. People make them much more complicated than they actually are. 

As Red points out, it's actually very simple: 



RedOnion said:


> The net is your interest rate (2.3%).



If you overpay a loan which charges you x% a year, you are saving x% a year. No more and no less.

Brendan


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## RedOnion (17 Dec 2018)

Ah sorry, I see it now (I think).

You're calculating the 6.2% as simple interest over 20 years.

Edit: I'm confusing myself at this stage! It's a coincidence that the simple interest return works out the same.


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## orka (17 Dec 2018)

SPC100 said:


> I don't think i have a good internal visualisation/representation of how my capital is included in the return. Can you explain it a bit more please?


Your 6%-ish return would be correct if the 10K was a normal investment and you got your 10K back after the 20years.  But you don't - it's gone.
It might help to imagine it as a one year 'investment'.  If someone offered you a 20% return (2,000) for one year on your 10K but they kept the 10K, you can probably see it's not a great deal despite the (interest/principal) looking high.  It's the same concept with the 20 years.


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## Zenith63 (17 Dec 2018)

Brendan Burgess said:


> People make them much more complicated than they actually are.


The calculations are logical and certainly since I've learned (through AAM) to focus on the interest it takes away much of the complexity, but for the average person their intuition is to factor capital payments in the 'cost' etc. - their intuition leads them astray or it is unintuitive...

Curious where I've gone wrong in this case coming to 1.2% though...


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## SPC100 (17 Dec 2018)

Brendan Burgess said:


> If you overpay a loan which charges you x% a year, you are saving x% a year. No more and no less.
> Brendan



So, based on wording like this, 10,000 overpaid on a loan of 2.3%,* one would expect a 230 p.a. saving.
*
BUT, in fact, the *outgoings actually reduce by 624.36 p.a.
*
I was surprised by the ~3 times difference in additional cashflow vs the "calculated saving".


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## RedOnion (17 Dec 2018)

SPC100 said:


> I was surprised by the ~3 times difference in additional cashflow vs the "calculated saving".


The  sum of all the differences between 230 and 624 is the initial 10,000 which you have left out of calculation.

If you give me 100 euro today, and I give you back 101 euro next year, you haven't made 101% return, you've made 1%.


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## SPC100 (17 Dec 2018)

RedOnion said:


> The  sum of all the differences between 230 and 624 is the initial 10,000 which you have left out of calculation.


it's not that simple...(624-230)*20 = 7880, or maybe I misunderstood you.


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## RedOnion (17 Dec 2018)

Deleted, as I explained so poorly


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## Brendan Burgess (17 Dec 2018)

I will say it again.

It's very simple. 

If you overpay a loan which charges you x% a year, you are saving x% a year. No more and no less.

If you come up with y, you are doing it wrong. 

Brendan


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## SPC100 (17 Dec 2018)

Brendan, while I understand the benefits of having rules/shortcuts, I like to understand the how, and be able to prove it to myself.



SPC100 said:


> So, based on wording like this, 10,000 overpaid on a loan of 2.3%,* one would expect a 230 p.a. saving.
> *
> BUT, in fact, the *outgoings actually reduce by 624.36 p.a. *



So, Why is the cash flow increase much higher than the calculated saving? I think I have internalised it now!

Three ways to think about this, that helped make it clearer:

1. If I pay down all 100k of the mortgage, I would no longer have to pay the annual mortgage payment of 6,240p.a. But my return can clearly not be 6.24% p.a. as the loan was by definition at 2.3%! i.e. My loan balance would go to 0, My future cashflow would increase by 6240, I would get a 2.3% financial return on my 100k.

2. Do a mortgage monthly calculation for the lump sum you are going to pay down using the term of your mortgage. e.g. 10,000, 20 years, 2.3% = 52.02 per month. i.e. If I pay down 10k, I will save 52.02 euro per month in future cashflow, or 52 euro per month is enough to pay a 10k 2.3% mortgage over twenty years. But the return is still 2.3% on your 10k.

3. Each mortgage payment is made up of some part to reduce the loan balance, and some part to pay off this months interest.

So, When I pay off the 10k, the loan balance instantly reduces by ten thousand (*I immediately have the benefit of the lower balance*), and I can only calculate my financial return as not paying interest on this 10k in the future. but my future payments will be lower as there is a) less capital to repay. b) lower interest amount due to lower balance on the loan.


*The financial return* is the same as your mortgage rate.
*The cashflow improvement*, is the same as what the lump sum would cost you to mortgage on the same terms as your mortgage.

I think is very interesting to keep the annual cashflow improvement in mind though when budgeting, and to be aware that it can be a lot larger than than lump sum multiplied by interest rate.

Thanks for the discussion folks, it helped me to figure it out.


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## Sarenco (17 Dec 2018)

Brendan Burgess said:


> If you overpay a loan which charges you x% a year, you are saving x% a year. No more and no less.


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).


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## JoeRoberts (17 Dec 2018)

To be pernickity, OP would also need to factor in the opportunity cost of not investing the extra repayments in another guaranteed savings product.


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## SPC100 (17 Dec 2018)

Otherways to look at at my return on my 10k.

If I invest 10k in a bank at 2.3% for 20 years. Ignoring tax, after twenty years I should have 10000*1.023^20 = 15758.

How do I show the same ~16k total return from my mortgage prepayment?

If I 'invest' 10k in a mortgage, I get a cashflow saving of 624 each year. 623*20 years = 12,480. But that is not accounting for any interest on each cashflow saving. If i can invest the 624 each year at 2.3%, my spreadsheet shows me making cashflow savings over the same 20 years of about 15,622, which is close enough to the hypothetical investment.


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## Brendan Burgess (17 Dec 2018)

So have you convinced yourself that your return is 2.3% ? 

Brendan


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## SPC100 (17 Dec 2018)

Yes. I am now convinced the return on my 10k prepayment is 2.3%.

But the annual cashflow improvement is not 230 euro (as one might assume from dealing with deposit accounts), but 624 euros.


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## SPC100 (17 Dec 2018)

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 (18 Dec 2018)

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.


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## SPC100 (18 Dec 2018)

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..

===answer===

*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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## Duke of Marmalade (18 Dec 2018)

Sarenco said:


> 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 (18 Dec 2018)

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


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## Duke of Marmalade (18 Dec 2018)

_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.


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## Brendan Burgess (18 Dec 2018)

Duke of Marmalade said:


> 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


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## Duke of Marmalade (18 Dec 2018)

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 (18 Dec 2018)

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:







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






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.


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## Sarenco (18 Dec 2018)

dub_nerd 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).


Yes, that's the point I was trying to make but no doubt expressed poorly.


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## Duke of Marmalade (18 Dec 2018)

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)_


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## Brendan Burgess (18 Dec 2018)

Duke of Marmalade said:


> 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


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## SPC100 (18 Dec 2018)

Duke of Marmalade said:


> _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.





dub_nerd said:


> 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 (18 Dec 2018)

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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## Duke of Marmalade (18 Dec 2018)

SPC100 said:


> 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).
> 
> ...


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.


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## SPC100 (18 Dec 2018)

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.


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## dub_nerd (19 Dec 2018)

Duke of Marmalade said:


> 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:






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






But if we then take this _c_ and hold it constant we can solve the first equation instead for "_N after_"






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:






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






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.


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## Duke of Marmalade (19 Dec 2018)

dub_nerd said:


> 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.


Good stuff, I love proof reading your posts.

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 (19 Dec 2018)

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:






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


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## Duke of Marmalade (19 Dec 2018)

dub_nerd said:


> 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:
> 
> 
> 
> ...


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.


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## dub_nerd (20 Dec 2018)

dub_nerd said:


>





Duke of Marmalade said:


> 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.



Could you help me out on what the reducing balance is, or how it is calculated? I can do simple maths but I haven't a financial bone in my body. I don't even understand your acronyms (IRR, CAGR?).


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## Duke of Marmalade (20 Dec 2018)

dub_nerd said:


> Could you help me out on what the reducing balance is, or how it is calculated? I can do simple maths but I haven't a financial bone in my body. I don't even understand your acronyms (IRR, CAGR?).


On a mortgage the reducing balance is the amount of mortgage outstanding after interest has been added and repayments have been deducted.  For regular repayments of c the RB is c(1 - (1+r)^(-Noutstanding))/r.  However, in general for more complicated cashflows on a loan the outstanding balance at any instant is the Net Present Value of future cashflows (NPV is an Excel function.)  It could also be calculated as the accumulation with interest of past cashflows.

IRR is the internal rate of return on a series of cashflows.  It is the discount rate which makes the discounted value of the cashflows zero.  It is an Excel function - play with it.  By definition the IRR of the cashflow vector with + M at time 0 and - c at each future time period is the mortgage rate.  It is a mathematical truism that any adjustment to the cashflows when analysed separately will also have an IRR equal to the mortgage rate.  This is simply because the lender will add mortgage interest to the outstanding balance at the end of each period.  IRR is a more appropriate term to use with a mixture of known positive and negative cash flows, and does not tell you whether you are a net investor or a net disinvestor.  Thus an IRR of 20% p.a. is very high but is only good news if on balance the timing of your cash flows is more negative in the early years.

*C*ompound *A*nnual *G*rowth *R*ate is a more appropriate term when one is clearly making an upfront investment with the hope of subsequent positive returns and where the concept of a predictable schedule of cashflows does not exist.  It is not a term which I use and was introduced by _Sarenco_.


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## dub_nerd (20 Dec 2018)

Ok, I think I understand the terms now. However, I don't know how to make it work so that the "return" on the lump sum is the same as the mortgage rate. Here's the decreasing mortgage balance for the 100k and 90k principals in the OP's example:






At all times 
	

	
	
		
		

		
		
	


	




  , the difference between the balances outstanding (i.e. the height of the blue strip) is equal to 
	

	
	
		
		

		
		
	


	




 . That is, we seem to be saving an amount equal to the value of the lump sum compounded at the mortgage interest rate. Originally I was happy to stop there and call it job done.

But as you pointed out, the blue strip between 
	

	
	
		
		

		
		
	


	




 and 
	

	
	
		
		

		
		
	


	




 includes an additional interest component that I have not accounted for. If I include this I don't get the mortgage interest rate whether I spread the savings over 
	

	
	
		
		

		
		
	


	




 or 
	

	
	
		
		

		
		
	


	




 periods (it's higher or lower than the mortgage rate, respectively). I presume this is where your "reducing balance" comes in, but I don't know what balance you are talking about or how to factor it in.


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## Duke of Marmalade (20 Dec 2018)

dub_nerd said:


> Ok, I think I understand the terms now. However, I don't know how to make it work so that the "return" on the lump sum is the same as the mortgage rate. Here's the decreasing mortgage balance for the 100k and 90k principals in the OP's example:
> 
> 
> 
> ...


Excellent. You have it completely right.  The calculation of the IRR is complicated and would need a spreadsheet, but by rational argument it must be the mortgage rate.  If you reproduce the change in cashflows on a spreadsheet and then apply the Excel IRR function you will get precisely the mortgage interest rate.  There are no simple formulas to produce the result directly.  Return on the lump sum is too loose a concept.  It is the return on the changed cashflow. That cashflow is a lump sum up front followed by savings on mortgage repayments between Na and N.


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## dub_nerd (20 Dec 2018)

Eureka! I finally understand it!

My attempt to simply spread the savings in interest evenly over _any_ given period was misguided (as you were trying to tell me). The correct way to do it is to treat the lump sum as an initial negative cash flow. Because we are keeping the constant repayment the same in both cases (with or without lump sum) there is no further net cash flow until time _Na_. Thereafter, there is a positive cash flow, equal to the constant payment per interval. The arrival of this cash flow over a period of time means that even though the payments are constant, their net present value isn't. An NPV calculation takes this into account using a discount rate. The related IRR calculation is a "backward" NPV calculation where we solve for the discount rate. It can only be done iteratively over all the periods -- there is no analytical solution, and certainly no naive interest rate calculation like I was trying.

Have I got that right?

This whole cashflow concept is new to me. When I bung the cashflows into Excel as described above, the IRR function returns the mortgage rate, correct to four significant figures. Very satisfying. 

... and all exactly as you said. 

Thanks for your patience.


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## dub_nerd (20 Dec 2018)

Addendum: ... and I also finally get SPC100's 'nit' at Post #23. If I take the positive cash flows as they arrive from time _Na_ onward, and invest them somewhere else at 2.3%, the final "simple" return will be 
	

	
	
		
		

		
		
	


	




  . I knew that had to crop up _somewhere_.


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## Duke of Marmalade (20 Dec 2018)

Yes _dub_ you have it spot on.  For someone who is not immersed in financial math as I have been throughout my actuarial career you have an amazing take on the issues.


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## dub_nerd (21 Dec 2018)




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## SPC100 (28 Dec 2018)

Thanks for the additional discussion on this thread dub_nerd and Duke, I enjoyed reading it, and it helped confirm my understanding of the return.


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## dub_nerd (28 Dec 2018)

After playing around a bit more I discovered that although Duke is correct about IRR not having a general analytic solution, for this _particular_ type of case it has a very neat one. Recalling that the cash flows are the initial payout of the lump sum at 
	

	
	
		
		

		
		
	


	




 , and then the constant repayment amount as a positive flow from month 
	

	
	
		
		

		
		
	


	




 onward, discounted by some percentage rate _r_, i.e.:






...we just need to show that this nets out to zero if we set _r_ equal to the mortgage rate. The sum term is a geometric series with common ratio 
	

	
	
		
		

		
		
	


	




, and initial term 
	

	
	
		
		

		
		
	


	




. The standard formula for the sum of a geometric series then gives:






With a bit of fiddling, this rearranges neatly to:






Now we recall from post 37 that if _r_ is the mortgage rate:






So by substituting for 
	

	
	
		
		

		
		
	


	




 we get:






Then we can rearrange the standard mortgage monthly payment formula:






... and using this substitution:






QED.


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## RichInSpirit (31 Dec 2018)

dub_nerd said:


> I can do simple maths but I haven't a financial bone in my body. I don't even understand your acronyms (IRR, CAGR?).



Your a mathematical genius Dub Nerd !
It's great to see compound interest formulas being laid bare on AAM. 
I know there are heaps of calculators around on the web but it's nice to be able to work stuff out for yourself.


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