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

#### dub_nerd

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

#### Duke of Marmalade

##### Frequent Poster
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.

Compound Annual Growth Rate 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

##### Frequent Poster
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
$n
, the difference between the balances outstanding (i.e. the height of the blue strip) is equal to
$L(1+r)^n$
. 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
$N_a$
and
$N$
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
$N_a$
or
$N$
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

##### Frequent Poster
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
$n
, the difference between the balances outstanding (i.e. the height of the blue strip) is equal to
$L(1+r)^n$
. That is, we seem to be saving an amount equal to the value of the lump sum compounded at the mortgage interest rate. But then the blue strip between
$N_a$
and
$N$
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
$N_a$
or
$N$
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.
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.

#### dub_nerd

##### Frequent Poster
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.

#### dub_nerd

##### Frequent Poster
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
$L(1+r)^N$
. I knew that had to crop up somewhere.

#### Duke of Marmalade

##### Frequent Poster
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.

#### SPC100

##### Frequent Poster
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.

#### dub_nerd

##### Frequent Poster
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
$n=0$
, and then the constant repayment amount as a positive flow from month
$N_a$
onward, discounted by some percentage rate r, i.e.:

$-L+\sum_{n=N_a}^{N}{\frac{c}{(1+r)^n}}$

...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
$(1+r)^{-1}$
, and initial term
$c(1+r)^{-N_a-1}$
. The standard formula for the sum of a geometric series then gives:

$\sum_{n=N_a}^{N}{\frac{c}{(1+r)^n}}=c(1+r)^{-N_a-1}\frac{1-(1+r)^{N-N_a-1}}{1-(1+r)^{-1}}$

With a bit of fiddling, this rearranges neatly to:

$\frac{c}{r}\left((1+r)^{-N_a}-(1+r)^{-N} \right )$

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

$N_a=\frac{\log\left(\frac{c}{c-r(P-L)} \right )}{\log(1+r)}=\log_{(1+r)}\left(\frac{c}{c-r(P-L)} \right )\Rightarrow(1+r)^{-N_a}=\frac{c-r(P-L)}{c}$

So by substituting for
$(1+r)^{-N_a}$
we get:

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

Then we can rearrange the standard mortgage monthly payment formula:

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

... and using this substitution:

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

QED.

#### RichInSpirit

##### Frequent Poster
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.